Unimodular mimetic $F(R)$ inflation

Abstract

We propose the unimodular-mimetic $F(R)$ gravity theory, to resolve cosmological constant problem and dark matter problem in a unified geometric manner. We demonstrate that such a theory naturally admits accelerating universe evolution. Furthermore, we construct unimodular-mimetic $F(R)$ inflationary cosmological scenarios compatible with the Planck and BICEP2/Keck-Array observational data. We also address the graceful exit issue, which is guaranteed by the existence of unstable de Sitter vacua.

Appendix: Explicit forms of the mimetic potential and Lagrange multipliers

Here we present the explicit forms of the mimetic potential and of the Lagrange multipliers for various cases appearing in the main text of the paper. We start off with the cosmological evolution (25), in which case the mimetic potential reads,

$$\begin{aligned}& V(t=\phi) \\& \quad= \bigl(27 t^{7} (1+w)^{5} (3+2 w) (4+3 w) (5+4 w) \\& \qquad{}\times (6+5 w) (7+6 w) (8+7 w) \bigr)^{-1} \\& \qquad{} -8 \bigl(-8 d (1-3 w)^{2} (3+2 w) (4+3 w) (5+4 w) \\& \qquad{}\times \bigl(45 t^{2} (1+w)^{2} (7+6 w) (8+7 w) \\& \qquad{}-3 t (6+5 w) (8+7 w) (9+7 w) \\& \qquad{} -2 (6+5 w) (7+6 w) (269+270 w) \bigr) \\& \qquad{}+t^{2} (1+w)^{2} (7+6 w) (8+7 w) \\& \qquad{}\times \bigl(9 t^{3} w (1+w)^{2} (4+3 w) \\& \qquad{}\times(5+4 w) (6+5 w)-2 f \bigl(-3+w (7+6 w) \bigr) \\& \qquad{}\times \bigl(-12 t (4+3 w) (6+5 w)+27 t^{2} (1+w)^{2} (5+4 w) \\& \qquad{}\times(6+5 w)-2 (4+3 w) (5+4 w) (107+108 w) \bigr) \bigr) \bigr), \\& \end{aligned}$$

(55)

and in addition, by using Eq. (57), the corresponding unimodular Lagrange multiplier function $\lambda(t)$ reads,

$$\begin{aligned} \lambda(t) =& \bigl(27 t^{7} (1+w)^{6} (3+2 w) (4+3 w) (5+4 w) \\ &{}\times(6+5 w) (7+6 w) (8+7 w) \bigr) \\ &{} - \bigl(8 \bigl(8 d (1-3 w)^{2} \bigl(60+133 w+98 w^{2}+24 w^{3} \bigr) \\ &{}\times \bigl(45 t^{2} (1+w)^{3} -2 \bigl(11298+41737 w+57679 w^{2} \\ &{}+35340 w^{3}+8100 w^{4} \bigr) \\ &{} +t \bigl(21216+104644 w+205616 w^{2}+201009 w^{3} \\ &{} +97725 w^{4}+18900 w^{5} \bigr)\bigr) \\ &{}\times t^{2} (1+w)^{2} \bigl(56+97 w+42 w^{2} \bigr) \\ &{}\times \bigl(9 t^{3} w (1+w)^{2} \bigl(240+692 w \\ &{}+740 w^{2}+347 w^{3}+60 w^{4} \bigr) f \bigl(-3+7 w+6 w^{2} \bigr) \\ &{}\times \bigl(27 t^{2} (1+w)^{3} \bigl(30+49 w+20 w^{2} \bigr) \\ &{}-2 \bigl(2140+7617 w \\ &{}+10109 w^{2}+5928 w^{3}+1296 w^{4} \bigr) \\ &{} +3 t \bigl(1224+5898 w+11595 w^{2}+11423 w^{3} \\ &{}+5586 w^{4}+1080 w^{5} \bigr) \bigr) \bigr) \bigr) \bigr) \end{aligned}$$

(56)

Correspondingly, by combining Eqs. (55) and (56), the mimetic Lagrange multiplier $\eta$ reads,

$$\begin{aligned} \eta(t) =& \bigl(27 t^{7} (1+w)^{6} (3+2 w) (4+3 w) (5+4 w) \\ &{}\times (6+5 w) (7+6 w) (8+7 w) \bigr)^{-1} \\ &{} - \bigl(4 \bigl(8 d (1-3 w)^{2} (3+2 w) (4+3 w) (5+4 w) \\ &{}\times\bigl(45 t^{2} (1+w)^{3} (7+6 w)(8+7 w) \\ &{}-2 (1+w) (6+5 w) (7+6 w) (269+270 w) \\ &{} +3 t (6+5 w) (8+7 w) \\ &{}\times \bigl(236+w \bigl(663+5 w (121+36 w) \bigr) \bigr) \bigr) \\ &{}\times t^{2} (1+w)^{2} (7+6 w) (8+7 w) \\ &{}\times \bigl(-9 t^{3} w (1+w)^{2} (4+3 w) \\ &{}\times(5+4 w) (6+5 w) (7+5 w) \\ &{}+2 f \bigl(-3+w (7+6 w) \bigr) \\ &{}\times \bigl(27 t^{2} (1+w)^{3} (5+4 w) (6+5 w) \\ &{} +6 t (4+3 w) (6+5 w) \\ &{}\times\bigl(58+w \bigl(161+w (137+36 w) \bigr) \bigr) \bigr) \bigr)\bigr)\bigr) \end{aligned}$$

(57)