-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathLegendre_Suppressed.nb
6723 lines (6642 loc) · 327 KB
/
Legendre_Suppressed.nb
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 11.3' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 328163, 6715]
NotebookOptionsPosition[ 321600, 6616]
NotebookOutlinePosition[ 322014, 6632]
CellTagsIndexPosition[ 321971, 6629]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell["Growth Index In F(R) Gravity.", "Title",
CellChangeTimes->{{3.825439480468724*^9, 3.8254395376088495`*^9},
3.8254456165835333`*^9, {3.827338567612325*^9,
3.827338579612709*^9}},ExpressionUUID->"d7ab949b-cfc1-4129-a464-\
cc7c8de1f955"],
Cell[CellGroupData[{
Cell["\<\
Enhancing/Dumping Gravitational Waves Using Viable \
F(R) Models Of Gravity\
\>", "Chapter",
CellChangeTimes->{{3.8254395534702497`*^9, 3.8254395776493025`*^9},
3.8254456192961254`*^9, {3.8254852471510305`*^9, 3.8254852611015496`*^9}, {
3.8273385960990233`*^9,
3.827338666277008*^9}},ExpressionUUID->"882ba6cb-0651-4719-b7a3-\
69596292c2a1"],
Cell[TextData[{
"Hereafter, MP is the Planck mass, ms is a mass scale related to ",
Cell[BoxData[
FormBox[
SuperscriptBox["\[Kappa]", "2"], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"1f07a8dd-4aa9-4fac-adb2-d07828c9e155"],
"\[Rho]0 with \[Rho]0 being the current density of dust, \[Chi] is the \
current density of radiation over dust, M is a different Mass scale, each \
function/parameter ending with \[OpenCurlyDoubleQuote]dot\
\[CloseCurlyDoubleQuote] implies derivative with respect to cosmic time t \
hence it is rewritten appropriately as a derivative of redshift z, \[Rho] is \
the sum of dust and radiation density, R0 is the real/proper Ricci scalar \
which is replaced in the f(R) function after calculations (for instance \
derivatives) have been carried, H2 stands for Hubble squared, it is easy to \
remember. YH is the function that we find as solution to the equations of \
motion instead of Hubble however yH is the proper statefinder we focus on, \
which is assigned the function satisfying the equations of motion, similar \
for the scalar field. Moreover, functions with the number 1 are similarly \
functions of the proper yH solution and facilitate us into extracting results \
for the statefinders. Referring to statefinders, q is as usual the \
deceleration parameter, j is the jerk and s is the snap, all of them \
connected with derivates of Hubble, however must be written with respect to \
redshift. Om is an extra statefinder for which mainly the current value \
matters, as it must coincide or actually be quite close to the current value \
of the matter density parameter \[CapitalOmega]m. Finally, functions ending \
with DE are related to dark energy, meaning all the extra geometric terms \
appearing in the equations of motion. In particular, \[Omega]DE stands for \
the Equations of state (EoS) parameter whereas \[CapitalOmega]DE as before \
signifies the Dark Energy density parameter. "
}], "Text",
CellChangeTimes->{{3.8254778959504457`*^9, 3.8254779109489126`*^9}, {
3.8254779463468113`*^9, 3.825478001745124*^9}, {3.8254793710855937`*^9,
3.825479409655266*^9}, {3.825479631003356*^9, 3.8254796596406736`*^9}, {
3.825489013049215*^9, 3.825489036937984*^9}, {3.825493230404848*^9,
3.8254932574634185`*^9}, {3.825493308076156*^9, 3.825493309181038*^9}, {
3.8255094092095585`*^9, 3.825509412769469*^9}, {3.827338682707926*^9,
3.8273387391552663`*^9}, {3.828955049635169*^9,
3.828955053546748*^9}},ExpressionUUID->"9f5cedc3-3654-41fe-8362-\
27e57f4bc654"],
Cell["", "Text",
CellChangeTimes->{{3.8254778959504457`*^9, 3.8254779109489126`*^9}, {
3.8254779463468113`*^9, 3.825478001745124*^9}, {3.8254793710855937`*^9,
3.825479409655266*^9}, {3.825479631003356*^9, 3.8254796596406736`*^9}, {
3.825489013049215*^9, 3.825489036937984*^9}, {3.825493230404848*^9,
3.8254932574634185`*^9}, {3.825493308076156*^9, 3.825493309181038*^9}, {
3.8255094092095585`*^9, 3.825509412769469*^9}, {3.827338682707926*^9,
3.827338739381098*^9}},ExpressionUUID->"5690f86d-e38a-423c-b80d-\
9bd7f696afff"],
Cell["\<\
In this notebook we are mainly interested in the effective equation for state \
EoS of the universe, denotes as weff[z]. Expected value is near 0, ideally it \
should be below 1/3 in order to be accurate and rises towards such value. \
Moreover we focus also on aMdz parameter, the sign of which suggests \
enhancement or dumping of primordial gravitational waves. The present model \
is capable of yielding interesting results as the effective EoS resides near \
0 for a while and afterwards starts increasing with a slow-rate converging on \
1/3. Inserting an auxiliary mass scale here denotes as R00 manages also to \
produce a finite but quite large ratio between the first 2 derivatives of the \
f(R). \
\>", "Text",
CellChangeTimes->{{3.8273387431478047`*^9, 3.827338867503422*^9}, {
3.8273389606680536`*^9, 3.8273390010598946`*^9}, {3.828955088003129*^9,
3.8289552255172634`*^9}},ExpressionUUID->"d9455074-64a0-4db0-9c85-\
707bd124e89b"],
Cell["\<\
\[CapitalLambda]CDM Values: Roughly speaking, q near -0.535, j near unity, \
snap near 0, Om near 0.315, \[Omega]DE near -1 (does not have to be exactly \
-1, it can also vary infinitesimally as shown in this particular model) and \
\[CapitalOmega]DE near 0.685 such stat Om+\[CapitalOmega]DE near unity \
(Friedman constraint). More details+accurate values can be found in arXiv: \
1807.06209.
\
\>", "Text",
CellChangeTimes->{{3.8254778959504457`*^9, 3.8254779109489126`*^9}, {
3.8254779463468113`*^9, 3.825478001745124*^9}, {3.8254793710855937`*^9,
3.825479409655266*^9}, {3.825479631003356*^9, 3.8254796596406736`*^9}, {
3.825489013049215*^9, 3.825489036937984*^9}, {3.825493230404848*^9,
3.8254932574634185`*^9}, {3.825493308076156*^9, 3.825493309181038*^9}, {
3.8255094092095585`*^9, 3.825509412769469*^9}, {3.827338682707926*^9,
3.8273387252325945`*^9}, {3.827338812093782*^9,
3.827338812893101*^9}},ExpressionUUID->"3b1d0326-8e65-49b3-9bba-\
6b367b8998df"],
Cell[CellGroupData[{
Cell["1. Designation Of Model.", "Section",
CellChangeTimes->{{3.8254395856770935`*^9, 3.8254395942811613`*^9}, {
3.8254399243942003`*^9,
3.82543992506089*^9}},ExpressionUUID->"9e6cfb22-6902-43db-bccf-\
c804edd60354"],
Cell[CellGroupData[{
Cell["1.1. Assigning Values To The Free Parameters.", "Subsection",
CellChangeTimes->{{3.82543993256345*^9, 3.825439947440945*^9}, {
3.825444376492654*^9,
3.8254443870757647`*^9}},ExpressionUUID->"cc920b98-15bf-4d9b-bace-\
ede6e8c9a438"],
Cell["\<\
Suppose we have a similar f(R) model as in arXiv:1912.13128 and \
arXiv:2003.06671 such that early-late time is unified. In order to solve \
numerically the equations of motion in the area [-0.9,1000] for redshift, one \
needs to specify the initial conditions, here denoted with 0 in the end and D \
in front for derivatives, given that we have second order derivatives for \
Hubble. Initial conditions for statefinder yH are taken from observations so \
they should not alter. In contrast, other parameters of the model, along with \
the initial conditions of the scalar field (since it is not observed) can be \
freely altered.\
\>", "Text",
CellChangeTimes->{{3.8254396093181753`*^9, 3.825439614643261*^9}, {
3.8254397218680615`*^9, 3.8254398956334476`*^9}, {3.8254439918573923`*^9,
3.8254440276604257`*^9}, {3.825478031245262*^9, 3.8254781829115047`*^9}, {
3.8254889600448556`*^9, 3.825488994304658*^9}, {3.8254934048110533`*^9,
3.825493407485289*^9}, {3.8254934407950096`*^9, 3.8254934422932515`*^9}, {
3.8273390293924994`*^9,
3.8273390803045936`*^9}},ExpressionUUID->"ffc6c881-04ba-4880-9c91-\
237055dcac52"],
Cell[BoxData[{
RowBox[{
RowBox[{"MP", "=",
RowBox[{"1.2", " ",
SuperscriptBox["10", "27"]}]}], ";"}], "\n",
RowBox[{
RowBox[{"ms", "=",
SqrtBox[
RowBox[{"1.87101", " ",
SuperscriptBox["10",
RowBox[{"-", "67"}]]}]]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"\[CapitalLambda]", "=",
RowBox[{"11.895", " ",
SuperscriptBox["10",
RowBox[{"-", "67"}]]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"Y0", "=",
RowBox[{
FractionBox["\[CapitalLambda]",
RowBox[{"3",
SuperscriptBox["ms", "2"]}]],
RowBox[{"(",
RowBox[{"1", "+",
FractionBox["11", "1000"]}], ")"}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"DY0", "=",
FractionBox["\[CapitalLambda]",
RowBox[{"3000",
SuperscriptBox["ms", "2"]}]]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"\[Chi]", "=",
RowBox[{"3.1", " ",
SuperscriptBox["10",
RowBox[{"-", "4"}]]}]}], ";"}]}], "Code",
CellChangeTimes->{{3.8254377907652693`*^9, 3.8254378400680604`*^9}, {
3.825438104488741*^9, 3.8254381248329835`*^9}, {3.8254385841996737`*^9,
3.825438590145526*^9}, {3.8254386875157623`*^9, 3.8254387358381834`*^9}, {
3.8254412473177137`*^9, 3.82544124765987*^9}, {3.8254439429597907`*^9,
3.825443950182592*^9}, {3.825444644456813*^9, 3.825444650680148*^9}, {
3.8254468746706066`*^9, 3.825446877832982*^9}, {3.825447294562168*^9,
3.8254473015364947`*^9}, {3.8254946517626925`*^9, 3.825494680920567*^9}, {
3.827337066796947*^9, 3.827337070896096*^9}, {3.8273404535021276`*^9,
3.827340457428815*^9}, {3.828777318691397*^9, 3.828777321091298*^9}, {
3.828953762294034*^9, 3.8289539956651554`*^9}, {3.8289552930179467`*^9,
3.8289552935909824`*^9}},
CellLabel->"In[1]:=",ExpressionUUID->"fc03874d-0924-4b95-8533-9fc0a24609ea"]
}, Open ]],
Cell[CellGroupData[{
Cell["1.2. Specifying The Model And The Equations Of Motion.", "Subsection",
CellChangeTimes->{{3.8254399600067215`*^9, 3.825439971652405*^9}, {
3.825444349787383*^9, 3.8254443676747723`*^9},
3.8274013150256805`*^9},ExpressionUUID->"8a201d8b-ba84-481d-8c6d-\
89383a43b5f3"],
Cell["\<\
Apart from defining the f(R) function, it is also useful to rewrite time \
derivatives as derivatives with respect to redshift z. Also, instead of \
Hubble\[CloseCurlyQuote]s parameter, we focus on a new statefinder function, \
denoted as YH in this framework in order to facilitate our study by \
simplifying a bit the equations of motion which were rendered intricate due \
to the variable change a(t)->z. The main focus lies with input eq1 as such \
differential equation shall be solved numerically in the aforementioned area \
for z. \
\>", "Text",
CellChangeTimes->{{3.825439983048732*^9, 3.825440193782028*^9}, {
3.825443090017993*^9, 3.825443273238162*^9}, {3.825478242695116*^9,
3.825478243894718*^9}, {3.825493471212572*^9, 3.825493472987229*^9}, {
3.8273390968790145`*^9, 3.827339135590562*^9}, {3.8274089108244815`*^9,
3.827408910825633*^9}},ExpressionUUID->"76dbbc44-6d60-4be5-b983-\
524082819074"],
Cell[TextData[{
"Note that changing the mass scale MP results in the appearance of \
singularities. I can reach recombination (z=3000) by making use of the Planck \
scale however higher scales can reach even further z without problems. Denote \
",
Cell[BoxData[
FormBox[
RowBox[{"R00", "=", " ",
RowBox[{
RowBox[{"(",
SuperscriptBox["10",
RowBox[{"-", "52"}]], ")"}], " ",
SuperscriptBox["ev", "2"]}]}], TraditionalForm]],
FormatType->"TraditionalForm",ExpressionUUID->
"33347398-8de6-4e5e-953a-2aa864c73fa6"],
", minimum finite value of the ratio F\[CloseCurlyQuote]\[CloseCurlyQuote]/F\
\[CloseCurlyQuote]. Think of it as a mass scale."
}], "Text",
CellChangeTimes->{{3.825439983048732*^9, 3.825440193782028*^9}, {
3.825443090017993*^9, 3.825443273238162*^9}, {3.825478242695116*^9,
3.825478243894718*^9}, {3.825493471212572*^9, 3.825493472987229*^9}, {
3.8273390968790145`*^9, 3.827339135590562*^9}, {3.827408913217312*^9,
3.827409043626974*^9}, {3.8278499447056837`*^9, 3.8278499757859497`*^9}, {
3.8278503331099005`*^9, 3.827850420736595*^9}, {3.8289548531444087`*^9,
3.8289549142896757`*^9}, {3.828954962009204*^9,
3.828954988788255*^9}},ExpressionUUID->"2cb94388-789a-4bac-80f8-\
9d8cca76782c"],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{"R00", "=",
SuperscriptBox["10",
RowBox[{"-", "52"}]]}], ";"}], "\n",
RowBox[{
RowBox[{
RowBox[{"f", "[", "R", "]"}], "=", " ",
RowBox[{"R", "+",
RowBox[{
RowBox[{"(",
RowBox[{"R", "+", "R00"}], ")"}], " ",
RowBox[{"LegendreP", "[",
RowBox[{"3", ",",
SuperscriptBox[
RowBox[{"(",
FractionBox[
RowBox[{"(",
RowBox[{"R", "+", "R00"}], ")"}],
SuperscriptBox["MP", "2"]], ")"}],
RowBox[{"-", "1"}]]}], "]"}]}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"fR", "[", "R", "]"}], "=",
RowBox[{"D", "[",
RowBox[{
RowBox[{"f", "[", "R", "]"}], ",", "R"}], "]"}]}], ";"}], "\n",
RowBox[{
RowBox[{
RowBox[{"fRdot", "[", "z", "]"}], "=",
RowBox[{
RowBox[{"Rdot", "[", "z", "]"}],
RowBox[{"D", "[",
RowBox[{
RowBox[{"fR", "[", "R", "]"}], ",", "R"}], "]"}]}]}], ";"}], "\n",
RowBox[{
RowBox[{
RowBox[{"\[Rho]", "[", "z", "]"}], "=",
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "z"}], ")"}], "3"],
RowBox[{"(",
RowBox[{"1", "+",
RowBox[{"\[Chi]",
RowBox[{"(",
RowBox[{"1", "+", "z"}], ")"}]}]}], ")"}]}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"H2", "[", "z", "]"}], "=",
RowBox[{
SuperscriptBox["ms", "2"],
RowBox[{"(",
RowBox[{
RowBox[{"YH", "[", "z", "]"}], "+",
RowBox[{"\[Rho]", "[", "z", "]"}]}], ")"}]}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"H", "[", "z", "]"}], "=",
SqrtBox[
RowBox[{"H2", "[", "z", "]"}]]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"R0", "[", "z", "]"}], "=",
RowBox[{
RowBox[{"12",
RowBox[{"H2", "[", "z", "]"}]}], "-",
RowBox[{"6",
RowBox[{"H", "[", "z", "]"}],
RowBox[{"(",
RowBox[{"1", "+", "z"}], ")"}],
RowBox[{"D", "[",
RowBox[{
RowBox[{"H", "[", "z", "]"}], ",", "z"}], "]"}]}]}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"Rdot", "[", "z", "]"}], "=",
RowBox[{
RowBox[{"-",
RowBox[{"H", "[", "z", "]"}]}],
RowBox[{"(",
RowBox[{"1", "+", "z"}], ")"}],
RowBox[{"D", "[",
RowBox[{
RowBox[{"R0", "[", "z", "]"}], ",", "z"}], "]"}]}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"eq1", "=",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"3",
RowBox[{"H2", "[", "z", "]"}],
RowBox[{"fR", "[", "R", "]"}]}], "-",
RowBox[{"3",
SuperscriptBox["ms", "2"],
RowBox[{"\[Rho]", "[", "z", "]"}]}], "+",
FractionBox[
RowBox[{"(",
RowBox[{
RowBox[{"f", "[", "R", "]"}], "-",
RowBox[{
RowBox[{"fR", "[", "R", "]"}],
RowBox[{"R0", "[", "z", "]"}]}]}], ")"}], "2"], "+",
RowBox[{"3",
RowBox[{"H", "[", "z", "]"}],
RowBox[{"fRdot", "[", "z", "]"}]}]}], ")"}], "/.",
RowBox[{"{",
RowBox[{"R", "\[Rule]",
RowBox[{"R0", "[", "z", "]"}]}], "}"}]}]}], ";"}], "\n",
RowBox[{
RowBox[{"A", "=",
RowBox[{
RowBox[{
RowBox[{"D", "[",
RowBox[{
RowBox[{"fR", "[", "R", "]"}], ",", "R"}], "]"}], "/",
RowBox[{"fR", "[", "R", "]"}]}], "//", "Simplify"}]}], ";"}], "\n",
RowBox[{"A", "/.",
RowBox[{"R", "\[Rule]", "0"}]}]}], "Code",
CellChangeTimes->{{3.825437874489356*^9, 3.825437938825757*^9}, {
3.8254379784505463`*^9, 3.82543823833449*^9}, {3.825438297868928*^9,
3.825438298118707*^9}, {3.825438367764273*^9, 3.8254385460278254`*^9}, {
3.825438634563583*^9, 3.825438635516983*^9}, {3.8254387847035866`*^9,
3.8254388765253987`*^9}, {3.8254412548671274`*^9, 3.825441257338137*^9},
3.825443930515047*^9, 3.8254446938017826`*^9, {3.825446781625005*^9,
3.825447018295049*^9}, {3.8254472688804393`*^9, 3.825447290654705*^9}, {
3.825477575269298*^9, 3.825477577919314*^9}, {3.8254947237321806`*^9,
3.8254947276716776`*^9}, {3.826460335084438*^9, 3.826460338357115*^9}, {
3.8273370741163826`*^9, 3.8273371373903694`*^9}, {3.8273372201497483`*^9,
3.8273372227160378`*^9}, {3.827340578898137*^9, 3.8273405791737213`*^9}, {
3.8273406768840513`*^9, 3.8273406939792495`*^9}, 3.8273407393343544`*^9, {
3.8273407740935364`*^9, 3.827340774740955*^9}, {3.8273408097888513`*^9,
3.8273408191505013`*^9}, {3.827340865438126*^9, 3.8273408721332045`*^9}, {
3.8273409395935497`*^9, 3.8273410491178303`*^9}, {3.827341085990063*^9,
3.827341258870195*^9}, {3.827399955610582*^9, 3.8274000379511404`*^9}, {
3.827400117766333*^9, 3.827400189380367*^9}, {3.827400248376565*^9,
3.827400269713344*^9}, {3.827400337594248*^9, 3.8274003380753565`*^9}, {
3.8274005120115957`*^9, 3.827400513266296*^9}, {3.827400656005763*^9,
3.8274006826033697`*^9}, 3.8274007251457157`*^9, {3.8274007905767193`*^9,
3.827400815994475*^9}, {3.8274008654786644`*^9, 3.8274009629293737`*^9}, {
3.8274010480433693`*^9, 3.8274010543091745`*^9}, {3.8274011116761403`*^9,
3.8274012559211*^9}, {3.827401446354593*^9, 3.8274014468999496`*^9}, {
3.8274014779974346`*^9, 3.827401488102919*^9}, {3.8274015215382423`*^9,
3.827401531140197*^9}, {3.827401598965688*^9, 3.8274016166105275`*^9}, {
3.827401748288105*^9, 3.8274017603270006`*^9}, {3.827401843444254*^9,
3.8274018441457167`*^9}, {3.8274018793531637`*^9,
3.8274019352869377`*^9}, {3.827402420873022*^9, 3.8274024793881865`*^9}, {
3.8274026175454407`*^9, 3.82740265205872*^9}, {3.8274027950138335`*^9,
3.8274028086474338`*^9}, {3.8274031539244227`*^9, 3.827403154801551*^9}, {
3.827403229832282*^9, 3.827403292434857*^9}, {3.827403398478969*^9,
3.8274036181389923`*^9}, {3.8274036839343805`*^9,
3.8274037150442843`*^9}, {3.827403755082877*^9, 3.827403773789056*^9}, {
3.8274038181978493`*^9, 3.827404000248584*^9}, {3.827404099688295*^9,
3.8274042105272026`*^9}, {3.827404259952738*^9, 3.827404284996716*^9}, {
3.827404320558341*^9, 3.827404449121451*^9}, {3.827404579897225*^9,
3.827404613475593*^9}, {3.8274046452057714`*^9, 3.827404754052128*^9}, {
3.827404813240399*^9, 3.827404869351022*^9}, {3.8274050085897274`*^9,
3.827405020905773*^9}, {3.827408300782585*^9, 3.827408320955776*^9}, {
3.8274083513988886`*^9, 3.8274084339752064`*^9}, {3.827408468212528*^9,
3.827408592080358*^9}, {3.82740862481094*^9, 3.8274086249412184`*^9}, {
3.8274086894311857`*^9, 3.827408712161847*^9}, {3.827408744520254*^9,
3.8274087517494917`*^9}, 3.827408846780365*^9, {3.8274089005109005`*^9,
3.8274089006772976`*^9}, {3.8278501755520263`*^9,
3.8278502155403795`*^9}, {3.8282717554293194`*^9, 3.828271756678665*^9}, {
3.828271929988799*^9, 3.82827193048416*^9}, {3.828703062674631*^9,
3.828703063054358*^9}, {3.828703207230923*^9, 3.8287032073624306`*^9}, {
3.8287033451291523`*^9, 3.8287034381681833`*^9}, {3.8287034781296997`*^9,
3.8287035563201904`*^9}, {3.8287035974302697`*^9,
3.8287036346499033`*^9}, {3.8287037104359784`*^9,
3.8287037111150007`*^9}, {3.828710610498893*^9, 3.828710634683983*^9}, {
3.8287106893330965`*^9, 3.828710850324901*^9}, {3.8287109638725615`*^9,
3.828711090845337*^9}, {3.828712725305505*^9, 3.828712725814624*^9}, {
3.828954004425131*^9, 3.8289541114015446`*^9}, 3.8289545699373245`*^9, {
3.8289549224514675`*^9, 3.828954958388794*^9}},
CellLabel->"In[7]:=",ExpressionUUID->"b7b61e00-6c4e-4d56-883e-cc63354f5054"],
Cell[BoxData[
RowBox[{"-", "3.`*^52"}]], "Output",
CellChangeTimes->{{3.8287033671521196`*^9, 3.828703439008266*^9}, {
3.828703479857043*^9, 3.828703556823491*^9}, {3.828703599948249*^9,
3.8287036350715027`*^9}, 3.82870371164753*^9, {3.8287106256253347`*^9,
3.8287106353879533`*^9}, {3.828710698221077*^9, 3.828710805561653*^9}, {
3.8287108396099663`*^9, 3.828710850770857*^9}, {3.8287109757703266`*^9,
3.828711091385764*^9}, 3.8287126333185263`*^9, 3.828712726665704*^9,
3.8287773366928787`*^9, {3.828954567821023*^9, 3.8289545704928284`*^9},
3.8289550024972334`*^9, 3.828955355262104*^9},
CellLabel->"Out[18]=",ExpressionUUID->"78689207-e0ef-4d4c-a7d7-66950c5c029f"]
}, Open ]],
Cell["\<\
Note: If I add another constant/scale, say R+ mass^2 in the input of the \
Legendre then the ratio F\[CloseCurlyQuote]\[CloseCurlyQuote]/F\
\[CloseCurlyQuote] at R->0 becomes finite and can tend to zero exactly as we \
want to. I would add a really small constant such that it is quite close to \
zero and no numerical difference is observed, but then the ratio is finite \
but quite large. Adding MP^2 results in an effectively zero ratio. It is best \
if I insert (R+scale^2) both in the Legendre and in front! The largest scale \
viable I observed that results in the smallest finite ratio is 10^(-52). The \
choice of constant I insert such that the ratio F\[CloseCurlyQuote]\
\[CloseCurlyQuote]/F\[CloseCurlyQuote] becomes finite affects the growth \
index, nothing happens if ms^2 is used but the ratio is quite large, but \
finite. The same applies to each model which used the Legendre formula. \
\>", "Text",
CellChangeTimes->{{3.8287037213048134`*^9, 3.8287038382068005`*^9}, {
3.828704995657405*^9, 3.8287050053203583`*^9}, {3.828705035971205*^9,
3.828705146662756*^9}, {3.828705177679371*^9, 3.8287051810959496`*^9}, {
3.8287055784441695`*^9, 3.828705600450364*^9}, {3.8287108947704854`*^9,
3.828710954759094*^9}, {3.8287111006919975`*^9,
3.8287111488203883`*^9}},ExpressionUUID->"fdecc3a3-f62f-461d-a346-\
434bb8946a78"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["2. Solving The Equation Of Motion", "Section",
CellChangeTimes->{{3.825440230114005*^9, 3.8254402419484253`*^9},
3.827340661230529*^9},ExpressionUUID->"05c864c2-2903-429e-8fff-\
674402c8fa46"],
Cell[CellGroupData[{
Cell["2.1. Derivation Of Statefinder yH .", "Subsection",
CellChangeTimes->{{3.825440248923175*^9, 3.8254402734843907`*^9},
3.8273406705633106`*^9},ExpressionUUID->"60ebff8e-0fcb-41af-9952-\
b502afaf398f"],
Cell["\<\
Using as input the initial conditions specified in section 1, the solution is \
extracted. Compatibility with Planck data is achieved if the current value of \
the statefinder yH is actually quite close to 2.\
\>", "Text",
CellChangeTimes->{{3.8254402861177654`*^9, 3.8254403157493715`*^9}, {
3.825493622933502*^9,
3.8254936883197393`*^9}},ExpressionUUID->"29c7dd68-d4ca-4b47-ba51-\
fa0dbc69eb7a"],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{"sol", "=",
RowBox[{"NDSolve", "[",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"eq1", "\[Equal]", "0"}], ",",
RowBox[{
RowBox[{"YH", "[", "10", "]"}], "\[Equal]", "Y0"}], ",",
RowBox[{
RowBox[{
RowBox[{"YH", "'"}], "[", "10", "]"}], "\[Equal]", "DY0"}]}], "}"}],
",",
RowBox[{"{", "YH", "}"}], ",",
RowBox[{"{",
RowBox[{"z", ",",
RowBox[{"-", "0.9"}], ",", "10000"}], "}"}]}],
"]"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"yH", "[", "z", "]"}], "=",
RowBox[{
RowBox[{"YH", "[", "z", "]"}], "/.",
RowBox[{"sol", "[",
RowBox[{"[", "1", "]"}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"yH", "[", "z", "]"}], "/.",
RowBox[{"z", "\[Rule]", "0"}]}], "//", "N"}]}], "Code",
CellChangeTimes->{{3.825438347276841*^9, 3.825438350521922*^9}, {
3.8254389812559032`*^9, 3.8254389954173183`*^9}, {3.8254439147470446`*^9,
3.825443916951807*^9}, {3.8273371580787315`*^9, 3.8273371859846544`*^9},
3.827338110314849*^9, 3.827339150391976*^9, {3.8274035485591636`*^9,
3.82740355041751*^9}, {3.827849849958635*^9, 3.827849850157527*^9}, {
3.827849981285452*^9, 3.8278499814127026`*^9}, {3.827850185606044*^9,
3.8278501857657447`*^9}, {3.8287030475231524`*^9,
3.8287030490242643`*^9}, {3.8287126387793427`*^9, 3.8287126392783175`*^9}},
CellLabel->"In[19]:=",ExpressionUUID->"75cb16db-4a6b-42bf-9bb4-891b57d43971"],
Cell[BoxData[
RowBox[{"{",
RowBox[{"{",
RowBox[{"YH", "\[Rule]",
InterpretationBox[
RowBox[{
TagBox["InterpolatingFunction",
"SummaryHead"], "[",
DynamicModuleBox[{Typeset`open$$ = False, Typeset`embedState$$ =
"Ready"},
TemplateBox[{PaneSelectorBox[{False -> GridBox[{{
PaneBox[
ButtonBox[
DynamicBox[
FEPrivate`FrontEndResource[
"FEBitmaps", "SquarePlusIconMedium"]],
ButtonFunction :> (Typeset`open$$ = True), Appearance ->
None, Evaluator -> Automatic, Method -> "Preemptive"],
Alignment -> {Center, Center}, ImageSize ->
Dynamic[{
Automatic, 3.5 CurrentValue["FontCapHeight"]/
AbsoluteCurrentValue[Magnification]}]],
GraphicsBox[{{{{}, {},
TagBox[{
Directive[
Opacity[1.],
RGBColor[0.368417, 0.506779, 0.709798],
AbsoluteThickness[1]],
LineBox[CompressedData["
1:eJwB0QMu/CFib1JlAgAAADwAAAACAAAAoWWKUWnL7L+6agP8gfQAQB9hSVg8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"]]},
Annotation[#,
"Charting`Private`Tag$3274#1"]& ]}}, {}, {}}, {
DisplayFunction -> Identity, Ticks -> {Automatic, Automatic},
AxesOrigin -> {0, 0}, FrameTicks -> {{{}, {}}, {{}, {}}},
GridLines -> {None, None}, DisplayFunction -> Identity,
PlotRangePadding -> {{
Scaled[0.1],
Scaled[0.1]}, {
Scaled[0.1],
Scaled[0.1]}}, PlotRangeClipping -> True, ImagePadding ->
All, DisplayFunction -> Identity, AspectRatio -> 1,
Axes -> {False, False}, AxesLabel -> {None, None},
AxesOrigin -> {0, 0}, DisplayFunction :> Identity,
Frame -> {{True, True}, {True, True}},
FrameLabel -> {{None, None}, {None, None}}, FrameStyle ->
Directive[
Opacity[0.5],
Thickness[Tiny],
RGBColor[0.368417, 0.506779, 0.709798]],
FrameTicks -> {{None, None}, {None, None}},
GridLines -> {None, None}, GridLinesStyle -> Directive[
GrayLevel[0.5, 0.4]], ImageSize ->
Dynamic[{
Automatic, 3.5 CurrentValue["FontCapHeight"]/
AbsoluteCurrentValue[Magnification]}],
Method -> {
"DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" ->
AbsolutePointSize[6], "ScalingFunctions" -> None,
"CoordinatesToolOptions" -> {"DisplayFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& ), "CopiedValueFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& )}},
PlotRange -> {{-0.9, 10000.}, {0., 23.313057828608937`}},
PlotRangeClipping -> True, PlotRangePadding -> {{
Scaled[0.1],
Scaled[0.1]}, {
Scaled[0.1],
Scaled[0.1]}}, Ticks -> {Automatic, Automatic}}],
GridBox[{{
RowBox[{
TagBox["\"Domain: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox[
RowBox[{"{",
RowBox[{"{",
RowBox[{
RowBox[{"-", "0.9`"}], ",", "10000.`"}], "}"}], "}"}],
"SummaryItem"]}]}, {
RowBox[{
TagBox["\"Output: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox["\"scalar\"", "SummaryItem"]}]}},
GridBoxAlignment -> {
"Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete ->
False, GridBoxItemSize -> {
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
GridBoxSpacings -> {
"Columns" -> {{2}}, "Rows" -> {{Automatic}}},
BaseStyle -> {
ShowStringCharacters -> False, NumberMarks -> False,
PrintPrecision -> 3, ShowSyntaxStyles -> False}]}},
GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False,
GridBoxItemSize -> {
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
BaselinePosition -> {1, 1}], True -> GridBox[{{
PaneBox[
ButtonBox[
DynamicBox[
FEPrivate`FrontEndResource[
"FEBitmaps", "SquareMinusIconMedium"]],
ButtonFunction :> (Typeset`open$$ = False), Appearance ->
None, Evaluator -> Automatic, Method -> "Preemptive"],
Alignment -> {Center, Center}, ImageSize ->
Dynamic[{
Automatic, 3.5 CurrentValue["FontCapHeight"]/
AbsoluteCurrentValue[Magnification]}]],
GraphicsBox[{{{{}, {},
TagBox[{
Directive[
Opacity[1.],
RGBColor[0.368417, 0.506779, 0.709798],
AbsoluteThickness[1]],
LineBox[CompressedData["
1:eJwB0QMu/CFib1JlAgAAADwAAAACAAAAoWWKUWnL7L+6agP8gfQAQB9hSVg8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"]]},
Annotation[#,
"Charting`Private`Tag$3274#1"]& ]}}, {}, {}}, {
DisplayFunction -> Identity, Ticks -> {Automatic, Automatic},
AxesOrigin -> {0, 0}, FrameTicks -> {{{}, {}}, {{}, {}}},
GridLines -> {None, None}, DisplayFunction -> Identity,
PlotRangePadding -> {{
Scaled[0.1],
Scaled[0.1]}, {
Scaled[0.1],
Scaled[0.1]}}, PlotRangeClipping -> True, ImagePadding ->
All, DisplayFunction -> Identity, AspectRatio -> 1,
Axes -> {False, False}, AxesLabel -> {None, None},
AxesOrigin -> {0, 0}, DisplayFunction :> Identity,
Frame -> {{True, True}, {True, True}},
FrameLabel -> {{None, None}, {None, None}}, FrameStyle ->
Directive[
Opacity[0.5],
Thickness[Tiny],
RGBColor[0.368417, 0.506779, 0.709798]],
FrameTicks -> {{None, None}, {None, None}},
GridLines -> {None, None}, GridLinesStyle -> Directive[
GrayLevel[0.5, 0.4]], ImageSize ->
Dynamic[{
Automatic, 3.5 CurrentValue["FontCapHeight"]/
AbsoluteCurrentValue[Magnification]}],
Method -> {
"DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" ->
AbsolutePointSize[6], "ScalingFunctions" -> None,
"CoordinatesToolOptions" -> {"DisplayFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& ), "CopiedValueFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& )}},
PlotRange -> {{-0.9, 10000.}, {0., 23.313057828608937`}},
PlotRangeClipping -> True, PlotRangePadding -> {{
Scaled[0.1],
Scaled[0.1]}, {
Scaled[0.1],
Scaled[0.1]}}, Ticks -> {Automatic, Automatic}}],
GridBox[{{
RowBox[{
TagBox["\"Domain: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox[
RowBox[{"{",
RowBox[{"{",
RowBox[{
RowBox[{"-", "0.9`"}], ",", "10000.`"}], "}"}], "}"}],
"SummaryItem"]}]}, {
RowBox[{
TagBox["\"Output: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox["\"scalar\"", "SummaryItem"]}]}, {
RowBox[{
TagBox["\"Order: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox["3", "SummaryItem"]}]}, {
RowBox[{
TagBox["\"Method: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox["\"Hermite\"", "SummaryItem"]}]}, {
RowBox[{
TagBox["\"Periodic: \"", "SummaryItemAnnotation"],
"\[InvisibleSpace]",
TagBox["False", "SummaryItem"]}]}},
GridBoxAlignment -> {
"Columns" -> {{Left}}, "Rows" -> {{Automatic}}}, AutoDelete ->
False, GridBoxItemSize -> {
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
GridBoxSpacings -> {
"Columns" -> {{2}}, "Rows" -> {{Automatic}}},
BaseStyle -> {
ShowStringCharacters -> False, NumberMarks -> False,
PrintPrecision -> 3, ShowSyntaxStyles -> False}]}},
GridBoxAlignment -> {"Rows" -> {{Top}}}, AutoDelete -> False,
GridBoxItemSize -> {
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
BaselinePosition -> {1, 1}]},
Dynamic[Typeset`open$$], ImageSize -> Automatic]},
"SummaryPanel"],
DynamicModuleValues:>{}], "]"}],
InterpolatingFunction[{{-0.9, 10000.}}, {
5, 7, 2, {25}, {4}, 0, 0, 0, 0, Automatic, {}, {},
False}, {{-0.9000000000000001, 0.18897352678813673`,
1.2779470535762734`, 2.367947053576273, 3.4579470535762735`,
4.547947053576274, 5.637947053576274, 6.7279470535762735`,
7.817947053576274, 8.907947053576274, 9.997947053576274,
9.998973526788136, 10., 10.210682354090478`, 10.421364708180954`,
1009.4213647081809`, 2008.421364708181, 3007.421364708181,
4006.421364708181, 5005.421364708181, 6004.421364708181,
7003.421364708181, 8002.421364708181, 9001.21068235409, 10000.}}, {
Developer`PackedArrayForm, {0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33,
36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72,
75}, CompressedData["
1:eJxTTMoPSmVkYGDwBuLMJ2dON35hcIhP95K0iUm0Z4CAhjDeFVutf2KKO5s1
z//6F1P8V3jI2g1MjBjiGytVd2ezY4pnzP52Qo0HU1x+7/FrDwUwxa/dm/Fk
jiimeC9D1qdwKUxxFyVrBmF5TPHfzjx855QxxY/aMBmdxyLeb/oj9AKmOMNT
p4lZu1Qwxa0DtOqXqGKKewVnC2zgEsQQL6tzfNjbJIkhfvEK903TvwoY4umv
yg6qWKpiiP/9/3ClcKkmhvhkUd9JTBt1McSnisbsrfhmgCGesGCiYYGIMYZ4
A6dJspeRKYb4gqJrU1QCzDHEATAFkWc=
"]}, {Automatic}],
Editable->False,
SelectWithContents->True,
Selectable->False]}], "}"}], "}"}]], "Output",
CellEditDuplicate->True,
CellChangeTimes->{
3.82543837131594*^9, 3.82543855102562*^9, 3.825438596189204*^9,
3.8254388904171095`*^9, 3.825438997311356*^9, 3.825477583613947*^9,
3.8254790964502664`*^9, 3.8273371865746384`*^9, 3.8273372252906437`*^9,
3.8273381125552626`*^9, 3.8273404649990387`*^9, 3.8273405822758055`*^9, {
3.8273406836700664`*^9, 3.827340697147923*^9}, 3.827340742617738*^9,
3.8273407777565565`*^9, {3.8273408127789316`*^9, 3.8273408219054728`*^9}, {
3.8273409308593836`*^9, 3.8273410524333696`*^9}, {3.8273410911528716`*^9,
3.827341230205177*^9}, 3.8273412626124024`*^9, {3.8273999679834275`*^9,
3.8274000408128967`*^9}, {3.8274001453804693`*^9, 3.8274001937833996`*^9},
3.827400251704105*^9, 3.827400317785952*^9, 3.827400394721586*^9,
3.827400515680518*^9, {3.8274006596400223`*^9, 3.8274006855007105`*^9},
3.8274007278466053`*^9, {3.8274007961798105`*^9, 3.8274008187539024`*^9}, {
3.8274008767573094`*^9, 3.8274009652251177`*^9}, {3.8274010506716986`*^9,
3.827401056670932*^9}, {3.827401155577407*^9, 3.8274012592224197`*^9}, {
3.8274014838485827`*^9, 3.8274014909946117`*^9}, {3.8274015250149636`*^9,
3.827401548191743*^9}, {3.8274016077661486`*^9, 3.827401619506832*^9}, {
3.8274017506527176`*^9, 3.8274017627443743`*^9}, 3.8274018475955496`*^9, {
3.827401883262439*^9, 3.8274019410540466`*^9}, 3.827402423700652*^9, {
3.8274024596098003`*^9, 3.8274024816441555`*^9}, {3.8274026199700994`*^9,
3.8274026546797028`*^9}, {3.827402801568084*^9, 3.827402810726883*^9},
3.827403161201578*^9, {3.82740325470103*^9, 3.8274032945361576`*^9}, {
3.827403401656573*^9, 3.827403620814305*^9}, {3.8274036892853966`*^9,
3.827403717429759*^9}, {3.82740375747398*^9, 3.8274039059749217`*^9}, {
3.827403961546508*^9, 3.827404004585336*^9}, {3.8274041051606727`*^9,
3.8274042127308226`*^9}, {3.827404263119276*^9, 3.8274042874729166`*^9}, {
3.8274043228058567`*^9, 3.827404362130085*^9}, {3.82740440890834*^9,
3.827404453320305*^9}, {3.8274045934614887`*^9, 3.8274046155396643`*^9}, {
3.827404647771823*^9, 3.82740475675121*^9}, {3.8274048238801165`*^9,
3.8274048804744067`*^9}, {3.8274050122581596`*^9, 3.827405023330415*^9}, {
3.827408305815894*^9, 3.8274083146613674`*^9}, {3.827408345585334*^9,
3.8274085949708805`*^9}, {3.827408682851759*^9, 3.8274087035888844`*^9}, {
3.8274087339661636`*^9, 3.827408754220048*^9}, 3.827408849650399*^9,
3.8274089031268287`*^9, 3.827409489291621*^9, 3.8275674149379025`*^9,
3.8278458570191946`*^9, 3.8278498602409573`*^9, 3.827849982194268*^9, {
3.827850187067279*^9, 3.8278502182696915`*^9}, {3.8282710121253505`*^9,
3.82827101730224*^9}, 3.8282717742577243`*^9, 3.828271933177836*^9,
3.828636879831579*^9, 3.8287030684540043`*^9, 3.8287032096474686`*^9,
3.8287034706695275`*^9, 3.8287035067270594`*^9, {3.828703536764138*^9,
3.8287035597559123`*^9}, 3.8287037158253183`*^9, 3.8287106410639*^9, {
3.8287107034282885`*^9, 3.828710807423624*^9}, {3.8287108430650644`*^9,
3.8287108523334436`*^9}, {3.828710979541258*^9, 3.8287110946005726`*^9},
3.8287126403530035`*^9, 3.828712730322592*^9, 3.828777342035789*^9,
3.8289545769604626`*^9, 3.828955355677947*^9},
CellLabel->"Out[19]=",ExpressionUUID->"48fe8b56-8362-4d67-bbb8-32099841f1d8"],
Cell[BoxData["2.121295450051042`"], "Output",
CellEditDuplicate->True,
CellChangeTimes->{
3.82543837131594*^9, 3.82543855102562*^9, 3.825438596189204*^9,
3.8254388904171095`*^9, 3.825438997311356*^9, 3.825477583613947*^9,
3.8254790964502664`*^9, 3.8273371865746384`*^9, 3.8273372252906437`*^9,
3.8273381125552626`*^9, 3.8273404649990387`*^9, 3.8273405822758055`*^9, {
3.8273406836700664`*^9, 3.827340697147923*^9}, 3.827340742617738*^9,
3.8273407777565565`*^9, {3.8273408127789316`*^9, 3.8273408219054728`*^9}, {
3.8273409308593836`*^9, 3.8273410524333696`*^9}, {3.8273410911528716`*^9,
3.827341230205177*^9}, 3.8273412626124024`*^9, {3.8273999679834275`*^9,
3.8274000408128967`*^9}, {3.8274001453804693`*^9, 3.8274001937833996`*^9},
3.827400251704105*^9, 3.827400317785952*^9, 3.827400394721586*^9,
3.827400515680518*^9, {3.8274006596400223`*^9, 3.8274006855007105`*^9},
3.8274007278466053`*^9, {3.8274007961798105`*^9, 3.8274008187539024`*^9}, {
3.8274008767573094`*^9, 3.8274009652251177`*^9}, {3.8274010506716986`*^9,
3.827401056670932*^9}, {3.827401155577407*^9, 3.8274012592224197`*^9}, {
3.8274014838485827`*^9, 3.8274014909946117`*^9}, {3.8274015250149636`*^9,
3.827401548191743*^9}, {3.8274016077661486`*^9, 3.827401619506832*^9}, {
3.8274017506527176`*^9, 3.8274017627443743`*^9}, 3.8274018475955496`*^9, {
3.827401883262439*^9, 3.8274019410540466`*^9}, 3.827402423700652*^9, {
3.8274024596098003`*^9, 3.8274024816441555`*^9}, {3.8274026199700994`*^9,
3.8274026546797028`*^9}, {3.827402801568084*^9, 3.827402810726883*^9},
3.827403161201578*^9, {3.82740325470103*^9, 3.8274032945361576`*^9}, {
3.827403401656573*^9, 3.827403620814305*^9}, {3.8274036892853966`*^9,
3.827403717429759*^9}, {3.82740375747398*^9, 3.8274039059749217`*^9}, {
3.827403961546508*^9, 3.827404004585336*^9}, {3.8274041051606727`*^9,
3.8274042127308226`*^9}, {3.827404263119276*^9, 3.8274042874729166`*^9}, {
3.8274043228058567`*^9, 3.827404362130085*^9}, {3.82740440890834*^9,
3.827404453320305*^9}, {3.8274045934614887`*^9, 3.8274046155396643`*^9}, {
3.827404647771823*^9, 3.82740475675121*^9}, {3.8274048238801165`*^9,
3.8274048804744067`*^9}, {3.8274050122581596`*^9, 3.827405023330415*^9}, {
3.827408305815894*^9, 3.8274083146613674`*^9}, {3.827408345585334*^9,
3.8274085949708805`*^9}, {3.827408682851759*^9, 3.8274087035888844`*^9}, {
3.8274087339661636`*^9, 3.827408754220048*^9}, 3.827408849650399*^9,
3.8274089031268287`*^9, 3.827409489291621*^9, 3.8275674149379025`*^9,
3.8278458570191946`*^9, 3.8278498602409573`*^9, 3.827849982194268*^9, {
3.827850187067279*^9, 3.8278502182696915`*^9}, {3.8282710121253505`*^9,
3.82827101730224*^9}, 3.8282717742577243`*^9, 3.828271933177836*^9,
3.828636879831579*^9, 3.8287030684540043`*^9, 3.8287032096474686`*^9,
3.8287034706695275`*^9, 3.8287035067270594`*^9, {3.828703536764138*^9,
3.8287035597559123`*^9}, 3.8287037158253183`*^9, 3.8287106410639*^9, {
3.8287107034282885`*^9, 3.828710807423624*^9}, {3.8287108430650644`*^9,
3.8287108523334436`*^9}, {3.828710979541258*^9, 3.8287110946005726`*^9},
3.8287126403530035`*^9, 3.828712730322592*^9, 3.828777342035789*^9,
3.8289545769604626`*^9, 3.8289553557666187`*^9},
CellLabel->"Out[21]=",ExpressionUUID->"718a7cfb-cb81-4f67-9605-ed2734485907"]
}, Open ]],
Cell["\<\
Afterwards, we assign the solutions in new functions and plot them. These \
will be used subsequently in order to study various statefinders and \
ascertain the validity of the model at hand. Furthermore, a comparison \
between the \[CapitalLambda]CDM Hubble and the one extracted in this model is \
also available.\
\>", "Text",
CellChangeTimes->{{3.82544036330894*^9, 3.8254403677769384`*^9}, {
3.825440408486622*^9, 3.825440442163652*^9}, {3.825440492555563*^9,
3.8254405198835745`*^9}, {3.8254471565971365`*^9, 3.825447202940959*^9}, {
3.8254937628189516`*^9, 3.8254938200001163`*^9}, {3.8273391710745173`*^9,
3.8273391722155094`*^9}},ExpressionUUID->"aa00ea59-ad04-4e84-8701-\
e79573874039"],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{"p1", "=",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"yH", "[", "z", "]"}], ",",
RowBox[{"{",
RowBox[{"z", ",",
RowBox[{"-", "0.9"}], ",", "100"}], "}"}], ",",
RowBox[{"Frame", "\[Rule]", "True"}], ",",
RowBox[{"FrameLabel", "\[Rule]",
RowBox[{"{",
RowBox[{"\"\<z\>\"", ",",
RowBox[{"Subscript", "[",
RowBox[{"\"\<y\>\"", ",", "\"\<H\>\""}], "]"}]}], "}"}]}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}], ",",
RowBox[{"PlotStyle", "\[Rule]", "Thick"}]}],
"]"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"H1", "[", "z", "]"}], "=",
RowBox[{"ms", " ",
SqrtBox[
RowBox[{
RowBox[{"yH", "[", "z", "]"}], "+",
RowBox[{"\[Rho]", "[", "z", "]"}]}]]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{"p2", "=",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"H1", "[", "z", "]"}], ",",
RowBox[{"{",
RowBox[{"z", ",",
RowBox[{"-", "0.9"}], ",", "20"}], "}"}], ",",
RowBox[{"Frame", "\[Rule]", "True"}], ",",
RowBox[{"FrameLabel", "\[Rule]",
RowBox[{"{",
RowBox[{"\"\<z\>\"", ",", "\"\<H\>\""}], "}"}]}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}], ",",
RowBox[{"PlotStyle", "\[Rule]",
RowBox[{"{",
RowBox[{"Green", ",", "Thick"}], "}"}]}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}]}], "\n",
RowBox[{
RowBox[{"H\[CapitalLambda]CDM0", "=",
RowBox[{"1.37187", " ",
SuperscriptBox["10",
RowBox[{"-", "33"}]]}]}], ";"}], "\n",
RowBox[{
RowBox[{"\[CapitalOmega]\[CapitalLambda]", "=", "0.6847"}], ";"}], "\n",
RowBox[{
RowBox[{"\[CapitalOmega]M", "=",
RowBox[{"0.3153", "-", "\[CapitalOmega]r"}]}], ";"}], "\n",
RowBox[{
RowBox[{"\[CapitalOmega]r", "=",
RowBox[{"8", " ",
SuperscriptBox["10",
RowBox[{"-", "5"}]]}]}], ";"}], "\n",
RowBox[{
RowBox[{
RowBox[{"H\[CapitalLambda]CDM", "[", "z", "]"}], "=",
RowBox[{"H\[CapitalLambda]CDM0", " ",
SqrtBox[
RowBox[{"\[CapitalOmega]\[CapitalLambda]", "+",
RowBox[{"\[CapitalOmega]M",
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "z"}], ")"}], "3"]}], "+",
RowBox[{"\[CapitalOmega]r",