Big bang nucleosynthesis constraints on scalar-tensor theories of gravity

We investigate Big bang nucleosynthesis (BBN) in scalar-tensor theories of gravity with arbitrary matter couplings and self-interaction potentials. We first consider the case of a massless dilaton with a quadratic coupling to matter. We perform a full numerical integration of the evolution of the scalar field and compute the resulting light element abundances. We demonstrate in detail the importance of particle mass thresholds on the evolution of the scalar field in a radiation dominated universe. We also consider the simplest extension of this model including a cosmological constant in either the Jordan or Einstein frame.

Figure 1

$a_{\mathrm{out}}$ as a function of $a_{\mathrm{in}}$ for different values of $\beta$ between 1 and 100. We see that $a_{\mathrm{out}}<a_{\mathrm{in}}$ which reflects the attraction towards general relativity during electron-positron annihilation.

Figure 2

The general structure of $a_{\mathrm{out}}$ as a function of $a_{\mathrm{in}}$ and $\beta$. This illustrates the complexity of the solutions of Eq. (44).

Figure 3

Evolution of the scalar field in phase space during a transition for $\beta =1$, 5, 50 from top to bottom when we assume ${\phi }_{*\mathrm{in}}=1$. The solid line corresponds to the exact solution of Eq. (54) while the dashed line corresponds to the solution of the approximate Eq. (56).

Figure 4

Evolution of $a_{\mathrm{out}}/a_{\mathrm{in}}$ as a function of $\beta$. Top: when we used the approximate Eq. (56), it does not depend on the initial value of the scalar field, ${\phi }_{*\mathrm{in}}$. (middle and bottom): we use the exact Eq. (54). This equation being nonlinear the ratio depends on the initial value of ${\phi }_{*\mathrm{in}}$.

Figure 5

The source function, $\Sigma (T)$, entering the Klein-Gordon equation when the mass thresholds corresponding to the particles listed in the text. The dashed curves show the individual particle contributions, ${\Sigma }_i(T)$, and the solid curve shows the sum, $\Sigma (T)$.

Figure 6

Dynamical evolution of ${\phi }_{*}$ when the source term is described by Fig. 5. (Top) we have set $\beta =1$ and ${\phi }_{*\mathrm{in}}=0.1$, 0.5, 1 (solid, dashed, dotted). (Bottom) We have fixed ${\phi }_{*\mathrm{in}}=1$ and taken $\beta =1$, 15, 20, 50 (thin solid, solid, dashed, dotted).

Figure 7

Evolution of $a_{\mathrm{ee}}/a_{\mathrm{in}}$ as a function of $\beta$ when the source term is described by Fig. 5 just before the electron-positron annihilation. Top: $\beta =1$. Bottom: $\beta =0.1$.

Figure 8

Comparison of the numerical integration and the analytical solution for a flat CDM model with $a_{\mathrm{in}}=1$.

Figure 9

${\sigma }_0$ as a function of ${\alpha }_{\mathrm{out}}$ and $\beta$. The labels on the contour lines give the value of $\log |{\sigma }_0|$.

Figure 10

Variation of the speed-up factor during BBN for various values of $\beta$ starting from the same value of $a_{\mathrm{in}}$. For small values of $\beta$, $\xi$ is almost constant and is driven toward 1 during the subsequent matter dominated era. For example, for $\beta \lesssim 0.2$ $\xi$ is constant during BBN. For larger values of $\beta$, the attraction toward general relativity is very efficient and $\xi \sim 1$ during BBN. The dashed line represents the time of $n/p$ freeze-out and we have indicated the interval during which light elements are formed.

Figure 11

$a({\varphi }_{*})$ as a function of $z$ for $\beta =10$ (top), $\beta =60$ (middle) and $\beta =1$ (bottom) assuming $a_{\mathrm{in}}=1$. The dashed lines show the redshift corresponding to matter-radiation equality. In the top panel, the dotted lines define $a_{\mathrm{in}}$, $a_{\mathrm{out}}$ and $a_0$ while in the middle panel it emphasizes the $(1+z{)}^{3/2}$ overall dependence of $a(\phi )$.

Figure 12

$\mathrm{D}$ mass fraction as a function of the baryonic density ($\log ({\eta }_{10})$) and $a_{\mathrm{in}}$ for $\beta =20$. The abundance of $\mathrm{D}$ is almost insensitive to $a_{\mathrm{in}}$.

Figure 13

Contour plots for $\mathrm{D}/\mathrm{H}$ as function of $a_{\mathrm{in}}$ and ${\eta }_{10}$ but for different values of $\beta$: 10, 20, 30 and 40. (For $\beta =20$, this is the same as Fig. 12.) The contours are evenly spaced by step of $\Delta \log (\mathrm{D}/\mathrm{H})=0.2$ starting from $\log (\mathrm{D}/\mathrm{H})=-4.8$ on the right. We conclude that the dependence of $\mathrm{D}$ mass fraction on $\beta$ is mild.

Figure 14

As in Fig. 12 but for ${}^4\mathrm{He}$. ${}^4\mathrm{He}$ is the most sensitive element to the value of $a_{\mathrm{in}}$.

Figure 15

As in Fig. 13 but for ${}^4\mathrm{He}$. Contours correspond from left to right to $Y_p=0.22$, 0.23, 0.24, 0.25, and 0.26 (i.e. black, red, green, blue and black, respectively.) The abundance of ${}^4\mathrm{He}$ is very sensitive to $\beta$. In addition, the dependence on $a_{\mathrm{in}}$ is increased for larger values of $\beta$.

Figure 16

As in Fig. 12 but for ${}^7\mathrm{Li}$. As in the case of deuterium, ${}^7\mathrm{Li}$ depends very mildly on $a_{\mathrm{in}}$.

Figure 17

As in Fig. 13 but for ${}^7\mathrm{Li}$. The contours are evenly spaced by steps of $\Delta \mathrm{Li}/\mathrm{H}={1.10}^{-10}$ starting from the value of $\mathrm{Li}/\mathrm{H}={1.10}^{-10}$ (in black only seen in the lower diagrams). ${}^7\mathrm{Li}$ shows little dependence on $\beta$ except for values of $\log ({\eta }_{10})$ corresponding to the minimum value of ${}^7\mathrm{Li}$.

Figure 18

${}^4\mathrm{He}$, $\mathrm{D}$ and ${}^7\mathrm{Li}$ abundances as a function of $a_{\mathrm{in}}$ for $\beta =5$, 10, 15, 20, 25, 30, 50, and 100 (The maximum deviations correspond to the lowest values of $\beta$). The baryonic density is set by WMAP observations and is assumed to be independent of the scalar component of gravity. For $\beta >10$ $\mathrm{D}$ is always compatible with observations and will set no constraint while ${}^7\mathrm{Li}$ cannot be reconciled with observations. We conclude that BBN constraints will mainly arise from ${}^4\mathrm{He}$.

Figure 19

Constraints on $({\alpha }_0,\beta )$ from ${}^4\mathrm{He}$ observations using WMAP measurement of ${\Omega }_{\mathrm{b}}$. The dashed line represents the constraint obtained in the solar system.

Figure 20

Same as Fig. 19 but with a cosmological constant either in Einstein frame or Jordan frame.

Figure 21

Evolution of the scalar field as a function of $z$ in models with a vanishing cosmological constant and a cosmological constant defined either in Einstein or Jordan frame. We see that only the late time dynamics is affected by the cosmological constant.