(* 1. Set Parameters *)

$Assumptions=k>0;$Assumptions=η>0;
aE=1;HI=1;Γ=0.014/(HI^6);c2[k_]:=-I c1[k];

(* 2. U0, U1 *)

(* FullSimplify[Expand[(solU1)-(U1[η,k] /. {c1[k]->c1,c2[k]->c2})]] *)

U0[η_,k_]:=c1[k](Sqrt[k η])BesselJ[0,k η]+c2[k](Sqrt[k η])BesselY[0,k η];
U1[η_,k_]:=-((288 (20727-4523 (Sqrt[21])))/300125)  ((c1[k] (HI^6))/((((((Sqrt[21])-3)/2)η aE HI))^(9+(Sqrt[21]))))(Sqrt[k η]) (2 π BesselY[0,k η] (-6 (1371162219+299210621 (Sqrt[21])) HypergeometricPFQ[{(1/2),-(9/2)-((Sqrt[21])/2)},{1,1,-(7/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-(3267117844+712937461 (Sqrt[21])) (k^2) (η^2) HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+7350 (1449+316 (Sqrt[21])) (k^4) (η^4) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+39 (147225227+32127118 (Sqrt[21])) (k^2) (η^2) HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-245 (714837+155983 (Sqrt[21])) (k^4) (η^4) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)])+(Sqrt[π]) BesselJ[0,k η] (78 (1705246067+372115053 (Sqrt[21])) k η MeijerG[{{(1/2),5+((Sqrt[21])/2)},{0}},{{(1/2),(1/2)},{-(1/2),-(1/2),(1/2) (8+(Sqrt[21]))}},k η,(1/2)]+7350 (13881+3029 (Sqrt[21])) (k^4) (η^4) MeijerG[{{(1/2),(1/2) (7+(Sqrt[21]))},{-(1/2)}},{{0,0},{-(1/2),0,(1/2) (5+(Sqrt[21]))}},k η,(1/2)]-1960 (1712457+373688 (Sqrt[21])) (k^3) (η^3) MeijerG[{{(1/2),(1/2) (8+(Sqrt[21]))},{0}},{{(1/2),(1/2)},{-(1/2),-(1/2),(1/2) (6+(Sqrt[21]))}},k η,(1/2)]-(37841511589+8257680071 (Sqrt[21])) (k^2) (η^2) MeijerG[{{(1/2),(1/2) (9+(Sqrt[21]))},{-(1/2)}},{{0,0},{-(1/2),0,(1/2) (7+(Sqrt[21]))}},k η,(1/2)]-72 (1551990251+338671484 (Sqrt[21])) MeijerG[{{(1/2),(1/2) (11+(Sqrt[21]))},{-(1/2)}},{{0,0},{-(1/2),0,(1/2) (9+(Sqrt[21]))}},k η,(1/2)])) +((288 (-852327+185993 (Sqrt[21])) (Sqrt[π]))/1500625)((c2[k] (HI^6))/(((((-3+(Sqrt[21]))/2)aE HI η))^(9+(Sqrt[21])))) (Sqrt[k η]) (BesselJ[0,k η] (7350 (119133+25997 (Sqrt[21]))(Sqrt[π]) (k^4) (η^4) HypergeometricPFQ[{(1/2),-(5/2)-((Sqrt[21])/2)},{1,1,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-980 (14697276+3207209 (Sqrt[21])) (Sqrt[π]) (k^4)  (η^4) HypergeometricPFQ[{(3/2),-(5/2)-((Sqrt[21])/2)},{2,2,-(3/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+(-268697011733-58634496497 (Sqrt[21]))(Sqrt[π]) (k^2)  (η^2) HypergeometricPFQ[{(1/2),-(7/2)-((Sqrt[21])/2)},{1,1,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]+39 (12108259609+2642238881 (Sqrt[21]))(Sqrt[π])(k^2) (η^2) HypergeometricPFQ[{(3/2),-(7/2)-((Sqrt[21])/2)},{2,2,-(5/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-24 (28192114587+6152023858 (Sqrt[21])) (Sqrt[π]) HypergeometricPFQ[{(1/2),-(9/2)-((Sqrt[21])/2)},{1,1,-(7/2)-((Sqrt[21])/2)},-(k^2) (η^2)]-14700 (570801+124559 (Sqrt[21])) (k^4) (η^4) MeijerG[{{(1/2) (7+(Sqrt[21]))},{(1/2),(1/2)}},{{0,0,0},{(1/2),(1/2) (5+(Sqrt[21]))}},k η,(1/2)]+1960 (140837769+30733321 (Sqrt[21])) (k^3) (η^3) MeijerG[{{(1/2) (8+(Sqrt[21]))},{0,0}},{{-(1/2),-(1/2),(1/2)},{0,(1/2) (6+(Sqrt[21]))}},k η,(1/2)]+4 (778050877142+169784621803 (Sqrt[21])) (k^2) (η^2) MeijerG[{{(1/2) (9+(Sqrt[21]))},{(1/2),(1/2)}},{{0,0,0},{(1/2),(1/2) (7+(Sqrt[21]))}},k η,(1/2)]-312 (35061208441+7650982944 (Sqrt[21])) k η MeijerG[{{5+((Sqrt[21])/2)},{0,0}},{{-(1/2),-(1/2),(1/2)},{0,(1/2) (8+(Sqrt[21]))}},k η,(1/2)]+72 (127640510767+27853443103 (Sqrt[21])) MeijerG[{{(1/2) (11+(Sqrt[21]))},{(1/2),(1/2)}},{{0,0,0},{(1/2),(1/2) (9+(Sqrt[21]))}},k η,(1/2)])+ BesselY[0,k η] (7350 (570801+124559 (Sqrt[21])) (k^4) (η^4) MeijerG[{{(1/2),(1/2) (7+(Sqrt[21]))},{-(1/2)}},{{0,0},{-(1/2),0,(1/2) (5+(Sqrt[21]))}},k η,(1/2)]-980 (140837769+30733321 (Sqrt[21])) (k^3) (η^3) MeijerG[{{0,(1/2) (8+(Sqrt[21]))},{-1}},{{-(1/2),(1/2)},{-1,-(1/2),(1/2) (6+(Sqrt[21]))}},k η,(1/2)]-2 (778050877142+169784621803 (Sqrt[21])) (k^2) (η^2) MeijerG[{{(1/2),(1/2) (9+(Sqrt[21]))},{-(1/2)}},{{0,0},{-(1/2),0,(1/2) (7+(Sqrt[21]))}},k η,(1/2)]+156 (35061208441+7650982944 (Sqrt[21])) k η MeijerG[{{0,5+((Sqrt[21])/2)},{-1}},{{-(1/2),(1/2)},{-1,-(1/2),(1/2) (8+(Sqrt[21]))}},k η,(1/2)]-36 (127640510767+27853443103 (Sqrt[21])) MeijerG[{{(1/2),(1/2) (11+(Sqrt[21]))},{-(1/2)}},{{0,0},{-(1/2),0,(1/2) (9+(Sqrt[21]))}},k η,(1/2)]));

(* 3. ψ0, ψ1 *)

ψ0[η_,k_]:=(aE^(((Sqrt[21])+1)/4))HI((((((Sqrt[21])-3)/2)aE HI η))^(-(1/2)))U0[η,k];
ψ1[η_,k_]:=(aE^(((Sqrt[21])+1)/4))HI((((((Sqrt[21])-3)/2)aE HI η))^(-(1/2)))(((13824 (477087-97732 (Sqrt[21])))/8575)((HI^6)/((((((Sqrt[21])-3)/2)η aE HI))^(9+(Sqrt[21]))))U0[η,k]+U1[η,k]);

(* 4. Power Spectrum *)

PS0[η_,k_]:=((((k/(aE HI))))^3)((Abs[ψ0[η,k]])^2);
PS[η_,k_]:=((((k/(aE HI))))^3)((Abs[ψ0[η,k]+Γ ψ1[η,k]])^2);

(* 5. Fig. 1 *)

Plot[Log[((PS[ η,(E^(-30))])/(PS[ 1.26,(E^(-30))]))],{η,1.26,1.8}]

Plot[{Log[((PS[ η,(E^(-40))])/(PS[ 1.26,(E^(-40))]))],Log[((PS[ η,(E^(-20))])/(PS[ 1.26,(E^(-20))]))],Log[((PS[ η,(E^(-10))])/(PS[ 1.26,(E^(-10))]))]},{η,1.26,1.8}]

(* 6. Fig.2 *)

aH[η_]:=aE HI(1/((((Sqrt[21])-3)/2)aE HI η))(1+((((13824 (477087-97732 (Sqrt[21])))/8575)Γ (HI^6))/((((((Sqrt[21])-3)/2)η aE HI))^(9+(Sqrt[21])))))Exp[-((1+(Sqrt[21]))/60)((((13824 (477087-97732 (Sqrt[21])))/8575)Γ (HI^6))/((((((Sqrt[21])-3)/2)η aE HI))^(9+(Sqrt[21]))))]; (* aH *)

Plot[Log[aH[η]],{η,1.3,1.5}]

logk=Table[Log[aH[1.3+((n-1)/200)]],{n,1,40}] (* log k *)

data=Table[{logk[[n]],Log[((PS[ 1.3+((n-1)/200),(E^(logk[[n]]))])/(PS[ 1.26,(E^(logk[[n]]))]))]},{n,1,40}]

ListPlot[data]

Fit[Take[data,10],{1,x},x]

1-0.062

Show[ListPlot[data],Plot[-3.0-0.062 x,{x,-36,-10}]]