Corrections to Newton’s law of gravitation in the context of codimension-1 warped thick braneworlds

In this work, we compute the corrections in Newton’s law of gravitation due to Kaluza-Klein gravitons in codimension-1 warped thick braneworld scenarios. We focus in some models recently proposed in the literature, the so-called asymmetric hybrid brane and compact brane. Such models are deformations of the ${\varphi }^4$ and sine-Gordon topological defects, respectively. Therefore we consider the branes engendered by such defects and we also compute the corrections in their cases. We use suitable numerical techniques to attain the mass spectrum and its corresponding eigenfunctions which are the essential quantities for computing the correction to the Newtonian potential. Moreover, we discuss that the existence of massive modes is necessary for building a braneworld model with a phenomenology involved. We find that the odd eigenfunctions have nontrivial contributions and the first eigenstate of the Kaluza-Klein tower has the highest contribution. The calculation of slight deviations in the gravitational potential may be used as a selection tool for braneworld scenarios matching with future experimental measurements in high energy collisions.

Figure 1

Potential $V_p(\varphi )$ for $p=1$ (thick line), $p=3$ (dashed line), $p=11$ (dotted line) and $p=63$ (thin line). The potential function acquires a compact shape in the $\varphi$-negative sector.

Figure 2

Scalar field solution ${\varphi }_p(y)$ in (a) and warp factor ${\sigma }_p(y)$ in (b) for different values of $p$. The asymmetric hybrid brane is engendered by a half-compacton defect which causes an asymmetry in the warp factor.

Figure 3

Scalar curvature for the asymmetric hybrid brane. The sudden change to a negative constant for $y\approx -0.5$ characterizes the hybrid behavior of the brane.

Figure 4

Plot of the potential in the curved space-time ${\mathcal{V}}_p$ for the same values of $p$ in Fig. 3. The potential acquires a limited shape in the negative part of the domain.

Figure 5

Analogue quantum potential for the asymmetric hybrid brane. The case for $p=1$ coincides with the ${\varphi }^4$ ordinary defect. As the hybrid profile evolves, the potential barrier diminishes.

Figure 6

Potential $V_{\lambda }(\varphi )$ for $\lambda =0.0$ (thick line), $\lambda =0.30$ (dashed line), $\lambda =0.50$ (dotted line) and $\lambda =0.65$ (thin line).

Figure 7

Scalar field solution ${\varphi }_{\lambda }(y)$ in (a) for different values of the parameter $\lambda$. Warp factor ${\sigma }_{\lambda }(y)$ in (b) for the same values of $\lambda$. As $\lambda$ increases, the warp factor becomes more and more compact. In the inset, we plot the warp factor in the compact regime ($\lambda =0.95$) and the Randall-Sundrum model with $k=30.5$.

Figure 8

Scalar curvature for the compact brane model for the same parameter values of Fig. 7. As the parameter $\lambda$ increases, the brane becomes more and more compressed around the origin. The inset shows the compact regime.

Figure 9

Plot of the potential in curved space-time ${\mathcal{V}}_{\lambda }(\varphi )$ for the same parameter values of Fig. 7. For $\lambda \to 0$, the potential coincides with the sine-Gordon model.

Figure 10

Analogue quantum potential $U_{\lambda }(\varphi )$ for $\lambda =0.0$ (thick line), $\lambda =0.30$ (dashed line), $\lambda =0.40$ (dotted line) and $\lambda =0.50$ (thin line). The case for $\lambda =0.0$ coincides with the sine-Gordon braneworld. The potential is very sensible in the parameter $\lambda$ and, as the compact profile arises, the potential barrier increases quickly. The inset corresponds to the scale adjustment of the $\lambda =0.95$ case.

Figure 11

Relative error in the approximation of the first mass eigenvalue of the Randall-Sundrum model in terms of the number of steps in the discretized domain.

Figure 12

Graviton mass spectrum for the asymmetric hybrid brane (squares) and compact brane (circles). The spectrum is model dependent and the whole sets decrease with the parameters. For the compact brane ($\lambda =0.96$), the spectrum is trivial.

Figure 13

Normalized squared odd wave functions for the asymmetric hybrid brane ($p=63$). The mass eigenvalues from Fig. 12 are $m_1=0.00804991$ (thick line), $m_3=0.0213199$ (dashed line) and $m_5=0.0344917$ (dotted line).

Figure 14

Normalized squared odd wave functions for the sine-Gordon brane ($\lambda =0.0$). All even wave functions vanish in $z=0$, so we omitted here. The mass eigenvalues from Fig. 12 are $m_1=0.0725202$ (thick line), $m_3=0.192295$ (dashed line) and $m_5=0.310824$ (dotted line).

Figure 15

Newtonian potential with corrections due to Kaluza-Klein modes (thick gray line). The asymmetric hybrid brane (dashed black line) has a negligible correction. The sine-Gordon brane (thick black line) provides a higher correction in comparison to the ordinary domain wall brane (thin black line).