1 Introduction

Although general relativity (GR) is the best theory of gravitational interaction and its predictions have been tested with high precision [1, 2], there are still open questions (arising at short distances and small time scales) for which GR predictability gets lost. Without any doubt, the prediction of the existence of the gravitational waves, or the CMBR as well as the formation of primordial light elements (Big Bang Nucleosynthesis) represent significant successes of GR. However, Einstein’s theory seems not to be completely satisfactory at the regime of the ultraviolet scale in which deviations from the Hilbert–Einstein action are needed. This leads to the introduction of new ingredients, such as dark matter and dark energy, necessary to fit the present picture of the Universe [3, 5,6,7,8,9,10,11,12]. A possibility that one may consider is the generalization of GR such that the gravitational action includes higher-order curvature invariants, \(\mathcal {L}\sim f(R,R_{\mu \nu }R^{\mu \nu },\Box ^k R,\dots )\) [13,14,15,16,17,18,19,20,21,22,23,24,25,26, 26,27,28,29,30,31]. The latter allows the inflationary behaviour of the early Universe to remove the primordial singularity and the explanation of the flatness and horizon problems [13, 14, 32,33,34,35,36,37,38,39,40,41,42,43]. Moreover, a high curvature regime requires the introduction of curvature invariants in order to build up self-consistent effective actions in curved spacetimeFootnote 1 [44,45,46]. Possible candidates for dark energy are a time-dependent cosmological term [52, 53], quintessence [54,55,56,57], dissipative fluids [58], Chaplygin gas [59, 60], K-essence [61,62,63,64], scalar–tensor theories [65,66,67,68,69,70,71], and other more exotic models [72]. Among the other possibilities to explain the accelerated expansion of the Universe, it has been also explored the possibility related to Brans–Dicke (BD) theory in 5D (without recurring to matter fields in 5D or dark energy in 4D). More specifically, it was shown in [73] (see also [66, 67, 74,75,76,77]) that the vacuum BD field equations in 5D are equivalent, on every hypersurface orthogonal to the extra dimension, to a BD theory in 4D equipped with a new matter–energy source of geometrical origin. This model implies that in a FRW cosmology, the reduced BD theory yields the accelerated expansion of a matter-dominated universe. This is consistent with current observations and with a decelerating radiation-dominated epoch. An extension of this model to f(R) gravity has been proposed in [78, 79]. This approach finds its bases in induced-matter theory. A formulation of 5D relativity, in which matter in 4D is counter-effect induced by geometry in 5D. In modified theories of gravity, such a scheme provides more degrees of freedom that can be exploited in order to build a coherent cosmological model according to observations.

In this work, we investigate how matter–antimatter asymmetry in the Universe and the formation of primordial light elements, that is the Big Bang Nucleosynthesis (BBN), can influence the framework descending from reduced BD 5D-cosmology to 4D. The origin of the baryon asymmetry in the early Universe is an unsolved problem of cosmology and particle physics. Observations show that the Universe is mostly made up of matter, contrary to what is expected from QFT in which matter and antimatter should be present in equal amounts. For such asymmetry to occur, the Sakharov conditions must be fulfilled [80]: (1) baryon number violation, (2) C and CP violation, and (3) deviation from thermal equilibrium. The CMB temperature anisotropies provide a strong probe of the baryon asymmetry. In fact, the observation of the acoustic peaks in CMB and the measurements of large-scale structures provide an estimation of the baryon asymmetry parameter \(\eta\) given by \(\eta ^{(CMB)}\sim (6.3\pm 0.3)\times 10^{-10}\) [81]. Yet, measurement of \(\eta\) can be also carried out in the context of the BBN, leading to \(\eta ^{(BBN)}\sim (3.4-6.9)\times 10^{-10}\) [82]. These two values are compatible, although derived in two different eras of the Universe evolution.

Moreover, we also explore the implications of the formation of light elements in the early Universe on 5D-cosmology. The latter occurred in the early phases of the Universe evolution, between the first fractions of a second after the Big Bang (\(\sim 0.01\) s) and a few hundred seconds after it, when the Universe was hot and dense (indeed BBN, together with cosmic microwave background radiation, provides the strong evidence about the high temperatures characterizing the primordial Universe). It describes the sequence of nuclear reactions that yielded the synthesis of light elements [83,84,85], and therefore drive the observed Universe. In general, from BBN physics, one may infer stringent constraints on a given cosmological model. In particular, in the present paper, after we have found some analytical solutions of field equations, we will derive the physical constraints on the free parameter \(\alpha\) of the model.

The paper is organized as follows. In section II, we recall the FRW cosmological equations inferred from the dimensional reduction of the scalar-vacuum BD field equations in 5D to 4D adopting the scheme of induced-matter theory. In section III, we study homogeneous and isotropic solutions of the vacuum BD field equations in 5D (focusing in particular to power-law solutions) applied to the baryon asymmetry in the Universe. We will discuss gravitational baryogenesis considering baryon asymmetry generated by the coupling of baryon current with the scalar Ricci curvature. In section IV, we study the consequences of 4D cosmological scenarios allowed by the 5D power-law solutions to the formation of light elements. In the last section, we draw our conclusions.

2 Dimensional reduction of Brans–Dicke theory in 5D

A Brans–Dicke (BD) theory of gravity in 5D is described by the action [73, 75]

$$\begin{aligned} S_{(5)} = \int {d^5 x \sqrt{|\gamma ^{(5)}|}\left[ \phi R^{(5)} - \frac{\omega }{\phi }\gamma ^{A B}\left( \nabla _{A}\phi \right) \left( \nabla _{B}\phi \right) \right] } + 16 \pi \int {d^5 x \sqrt{|\gamma ^{(5)}|}L_{m}^{(5)}}, \end{aligned}$$
(1)

where \(R^{(5)}\) is the curvature scalar associated with the 5D metric \(\gamma _{A B}\); \(\gamma ^{(5)}\) is the determinant of \(\gamma _{A B}\); \(\phi\) is a scalar field; \(\omega\) is a dimensionless coupling constant; and \(L_{m}^{(5)}\) represents the Lagrangian of the matter fields in 5D and does not depend on \(\phi\). The effective equations for gravity in 4D are given byFootnote 2

$$\begin{aligned} G_{\mu \nu }^{(4)} = \frac{8\pi }{\phi }\left( S_{\mu \nu } + T_{\mu \nu }^{(BD)}\right) + \frac{\omega }{\phi ^2}\left[ \left( D_{\mu }\phi \right) \left( D_{\nu }\phi \right) - \frac{1}{2}g_{\mu \nu } \left( D_{\alpha }\phi \right) \left( D^{\alpha }\phi \right) \right] + \frac{1}{\phi }\left( D_{\mu }D_{\nu }\phi - g_{\mu \nu } D^2\phi \right) - g_{\mu \nu }\frac{V(\phi )}{2 \phi } \end{aligned}$$
(2)

where we have introduced the quantity \(V(\phi )\), which (as we will see bellow) plays the role of an effective or induced scalar potential; \(S_{\mu \nu }\) is the reduced energy-momentum tensor (EMT) of the matter fields in 5D

$$\begin{aligned} S_{\mu \nu } = T_{\mu \nu }^{(5)} - g_{\mu \nu }\left[ \frac{\left( \omega + 1\right) T^{(5)}}{4 + 3 \omega } - \frac{\epsilon T_{44}^{(5)}}{\Phi ^2}\right] , \end{aligned}$$
(3)

while \(T_{\mu \nu }^{(BD)}\) can be interpreted as an induced energy-momentum tensor for an effective BD theory in 4D, given by [73]

$$\begin{aligned} 8\pi T_{\mu \nu }^{(BD)} = 8 \pi T_{\mu \nu }^{(STM)} + \frac{\epsilon {\mathop {\phi }\limits ^{*}}}{2 \Phi ^2 }\left[ {\mathop {g}\limits ^{*}}_{\mu \nu } + g_{\mu \nu }\left( \frac{\omega {\mathop {\phi }\limits ^{*}}}{\phi } - g^{\alpha \beta }{\mathop {g}\limits ^{*}}_{\alpha \beta }\right) \right] + \frac{1}{2}g_{\mu \nu }V, \end{aligned}$$
(4)

with

$$\begin{aligned} \frac{8\pi }{\phi }T_{\mu \nu }^{(STM)}\equiv & {} \frac{D_{\mu }D_{\nu }\Phi }{\Phi } - \frac{\epsilon }{2\Phi ^2}\left\{ \frac{{\mathop {\Phi }\limits ^{*}}{\mathop {g}\limits ^{*}}_{\mu \nu }}{\Phi } - {\mathop {g}\limits ^{**}}_{\mu \nu } + g^{\alpha \beta }{\mathop {g}\limits ^{*}}_{\mu \alpha }{\mathop {g}\limits ^{*}}_{\nu \beta } - \frac{g^{\alpha \beta }{\mathop {g}\limits ^{*}}_{\alpha \beta }{\mathop {g}\limits ^{*}}_{\mu \nu }}{2} + \frac{g_{\mu \nu }}{4}\left[ {\mathop {g}\limits ^{*}}^{\alpha \beta }{\mathop {g}\limits ^{*}}_{\alpha \beta } + \left( g^{\alpha \beta }{\mathop {g}\limits ^{*}}_{\alpha \beta }\right) ^2\right] \right\} . \end{aligned}$$
(5)

Since in BD, \(\phi\) acts as the inverse of the Newtonian gravitational constant G, (5) is identical to the induced EMT used in STM (space–time–matter theory) [86]. The second term in (4) depends on the first derivatives of \(\phi\) with respect to the fifth coordinate and represents the effective EMT in 4D coming from the scalar field. Taking the trace of (2), we obtain a simple relation between \(R^{(4)}\), \(S = g^{\mu \nu }S_{\mu \nu }\) and \(T^{(BD)} = g^{\mu \nu } T_{\mu \nu }^{(BD)}\), namely (we note that \(g^{\mu \nu }T_{\mu \nu }^{(5)} = T^{(5)} - \epsilon T_{44}^{(4)}/\Phi ^2\))

$$\begin{aligned} R^{(4)} = - \frac{8\pi }{\phi }\left( S + T^{(BD)}\right) + \frac{\omega \left( D_{\alpha }\phi \right) \left( D^{\alpha }\phi \right) }{\phi ^2} + \frac{3 D^2 \phi }{\phi } + \frac{2 V}{\phi }. \end{aligned}$$
(6)

Since we are interested in induced-matter solutions of 5D Brans–Dicke theory hereon, we will focus on a scalar-vacuum Brans–Dicke cosmology in 5D. Actually, a general key point for the embedding of 4D Einstein theory into a 5D Kaluza–Klein induced-matter approach is provided by Campbell–Magaard (CM) theorem [87]. This theorem states that any analytic Riemannian space of N-dimension can be locally embedded in a Ricci-flat Riemannian space of \((N+1)\)-dimension. As a consequence, the 5D metric (A5) is commonly taken, in cosmological applications, considering 4D spacetime as a hypersurface \(y\,=\,y_0\,=const\) orthogonal to the 5D extra-coordinate. Therefore one can consider the line element in the form:

$$\begin{aligned} ds^2 = n^2(t, y)dt^2 - a^2(t, y)\left[ \frac{dr^2}{1 - k r^2} + r^2 \left( d\theta ^2 + \sin ^2 \theta d\varphi ^2\right) \right] + \epsilon \Phi ^2(t, y)dy^2, \end{aligned}$$
(7)

where \(k = 0, + 1, - 1\) and \((t, r, \theta , \phi )\) are the usual coordinates for a spacetime with spherically symmetric spatial sections. If one considers the previous metric in field equations, the vacuum \((T_{AB}^{(5)} = 0)\) Brans–Dicke cosmological equations (see (A1)) give: for the temporal component \(A = B = 0\)

$$\begin{aligned} 3\frac{\dot{a}}{a}\left( \frac{{\dot{a}}}{a} + \frac{\dot{\Phi }}{ \Phi }\right) + \frac{3 k n^2}{a^2} +\frac{3 \epsilon n^2}{\Phi ^2}\left[ \frac{{\mathop {a}\limits ^{**}}}{a} + \frac{{\mathop {a}\limits ^{*}}}{a}\left( \frac{{\mathop {a}\limits ^{*}}}{a} - \frac{{\mathop {\Phi }\limits ^{*}}}{\Phi }\right) \right] = \frac{1}{\phi }\left[ \ddot{\phi } + \dot{\phi }\left( \frac{\omega \dot{\phi }}{2 \phi } - \frac{\dot{n}}{n}\right) \right] + \frac{\epsilon n^2 {\mathop {\phi }\limits ^{*}}}{\Phi ^2 \phi }\left( \frac{{\mathop {n}\limits ^{*}}}{n} - \frac{\omega {\mathop {\phi }\limits ^{*}}}{2 \phi }\right) \end{aligned}$$
(8)

while, for spatial components \(A = B = 1, 2, 3\)

$$\begin{aligned}& \frac{2\ddot{a}}{a} + \frac{\dot{a}}{a}\left( \frac{\dot{a}}{a} - \frac{2\dot{n}}{n}\right) + \frac{\ddot{\Phi }}{\Phi } + \frac{\dot{\Phi }}{\Phi }\left( \frac{2\dot{a}}{a} - \frac{\dot{n}}{n}\right) + \frac{k n^2}{a^2} +\frac{\epsilon n^2}{\Phi ^2}\left[ \frac{2{\mathop {a}\limits ^{**}}}{a} + \frac{{\mathop {a}\limits ^{*}}}{a}\left( \frac{{\mathop {a}\limits ^{*}}}{a} + \frac{2 {\mathop {n}\limits ^{*}}}{n}\right) + \frac{{\mathop {n}\limits ^{**}}}{n} - \frac{{\mathop {\Phi }\limits ^{*}}}{\Phi }\left( \frac{2{\mathop {a}\limits ^{*}}}{a} + \frac{{\mathop {n}\limits ^{*}}}{n}\right) \right] \nonumber \\ & \quad = \frac{\dot{\phi }}{ \phi }\left( \frac{\dot{a}}{a} - \frac{\omega \dot{\phi }}{2 \phi }\right) + \frac{\epsilon n^2 {\mathop {\phi }\limits ^{*}}}{\Phi ^2 \phi }\left( \frac{{\mathop {a}\limits ^{*}}}{a} - \frac{\omega {\mathop {\phi }\limits ^{*}}}{2 \phi }\right) , \end{aligned}$$
(9)

along the extra-coordinate \(A = B = 4\)

$$\begin{aligned} 3\left[ \frac{\ddot{a}}{a} + \frac{\dot{a}}{a}\left( \frac{\dot{a}}{a} - \frac{\dot{n}}{n}\right) \right] + \frac{3 k n^2}{a^2} + \frac{3 \epsilon n^2}{\Phi ^2 }\left( \frac{{\mathop {a}\limits ^{*}}}{a}\right) \left( \frac{{\mathop {a}\limits ^{*}}}{a} + \frac{{\mathop {n}\limits ^{*}}}{n}\right) = \frac{\dot{\phi }}{\phi }\left( \frac{\dot{\Phi }}{\Phi } - \frac{\omega \dot{\phi }}{2 \phi }\right) + \frac{\epsilon n^2}{\Phi ^2 \phi } \left[ {\mathop {\phi }\limits ^{**}}+ {\mathop {\phi }\limits ^{*}}\left( \frac{\omega {\mathop {\phi }\limits ^{*}}}{2 \phi } - \frac{{\mathop {\Phi }\limits ^{*}}}{\Phi }\right) \right] , \end{aligned}$$
(10)

and, finally, for the mixed component \(A = 0, B = 4\)

$$\begin{aligned} 3\left( \frac{{\mathop {n}\limits ^{*}} \dot{a}}{n a} + \frac{\dot{\Phi }{\mathop {a}\limits ^{*}}}{\Phi a} - \frac{{\mathop {\dot{a}}\limits ^{*}}}{a}\right) = \frac{{\mathop {\dot{\phi }}\limits ^{*}}}{\phi } - \frac{{\mathop {n}\limits ^{*}} \dot{\phi }}{n \phi } - \frac{\dot{\Phi } {\mathop {\phi }\limits ^{*}}}{\Phi \phi } + \frac{\omega }{\phi ^2}\dot{\phi }{\mathop {\phi }\limits ^{*}}\,. \end{aligned}$$
(11)

The wave equation (A4) reads

$$\begin{aligned} \nabla ^2 \phi = \frac{1}{n^2}\left[ \ddot{\phi } + \dot{\phi }\left( \frac{3 \dot{a}}{a} + \frac{\dot{\Phi }}{\Phi } - \frac{\dot{n}}{n}\right) \right] + \frac{\epsilon }{\Phi ^2}\left[ {\mathop {\phi }\limits ^{**}}+ {\mathop {\phi }\limits ^{*}}\left( \frac{3 {\mathop {a}\limits ^{*}}}{a} + \frac{{\mathop {n}\limits ^{*}}}{n} - \frac{{\mathop {\Phi }\limits ^{*}}}{\Phi }\right) \right] = 0. \end{aligned}$$
(12)

A suitable solution set for these equations can be inferred by assuming that the metric coefficients are separable functions of their arguments [73]

$$\begin{aligned} n(t, y) = N(y),\;\;\; a(t, y) = P(y)Q(t),\;\;\; \Phi (t, y) = F(t),\;\;\; \phi (t, y) = U(y)W(t). \end{aligned}$$
(13)

In particular, in the hypothesis of a spatially-flat scenario, as it is suggested by observations, one may consider power-law solutions

$$\begin{aligned} n(t, y) = A y^{\alpha }, \;\;\;a(t, y)= B y^{\beta }t^{l}, \;\;\;\Phi (t, y) = C t^{m}, \;\;\;\phi (t, y) = D y^{\gamma }t^s. \end{aligned}$$
(14)

Here A, B, C, and D are some constants with the appropriate units; while \(\alpha , \beta , \gamma\) while l, m, and s are parameters that will be constrained by field Eqs. (8)–(12). In particular, substituting (14) into (12), we obtain

$$\begin{aligned} s\left( s - 1 + 3 l + m\right) C^2 t^{(s - 2)}y^{(\gamma - 2\alpha )} + \epsilon \gamma \left( \gamma - 1 + 3\beta + \alpha \right) A^2 t^{(s - 2 m)}y^{(\gamma - 2)} = 0. \end{aligned}$$
(15)

This equation can be satisfied with several sets of parameters (a detailed analysis of these solutions can be found in [73]):

  • \(s = 0, \gamma = 0;\)

  • \(s = 0, \gamma = 1 - 3\beta - \alpha ;\)

  • \(\gamma = 0,s = 1 - 3 l - m;\)

  • \(s = 1 - 3 l - m, \gamma = 1 - 3 \beta - \alpha ;\)

  • \(m = 1, \alpha = 1, \;\;\;s \left( s + 3\,l\right) C^2 + \epsilon \gamma \left( \gamma + 3\beta \right) A^2 = 0.\)

Here, we focus on the case \(V = 0\) (Sect. IV of [73]) that implies

$$\begin{aligned} \gamma= & {} \frac{s\left( s + 6 l - 2\right) \left( s + 3 l\right) }{\left( s + 3 l - 1\right) \left[ s^2 + s\left( 3 l - 2\right) + 6 l\left( l - 1\right) \right] }. \end{aligned}$$
(16)
$$\begin{aligned} \omega= & {} \frac{6\left( 2 l - 1\right) l}{s\left( s + 6 l - 2\right) }, \quad p = n \rho , \quad n = \frac{2 - s - 3l}{3 l}, \quad s = \frac{2\left( 2 l - 1\right) }{\left( 1 - n\right) \left[ 2 - 3 l\left( n + 1\right) \right] }. \end{aligned}$$
(17)

Consequently, a effective BD cosmology in 4D can be developed in terms of the only metric potentials a(t) and \(\phi (t)\) written as

$$\begin{aligned} a(t)= & {} a_{f} \left( \frac{t}{t_{f}}\right) ^{\alpha }, \quad \alpha = \frac{2\left[ 1 + \omega \left( 1 - n\right) \right] }{\left[ 4 + 3\omega \left( 1 - n^2\right) \right] } \end{aligned}$$
(18)
$$\begin{aligned} \phi (t)= & {} \phi _{f}\left( \frac{t}{t_{f}}\right) ^{\beta },\quad \beta = \frac{2\left( 1 - 3 n\right) }{\left[ 4 + 3\omega \left( 1 - n^2\right) \right] } , \quad \phi _{f} = \frac{4\pi \rho _{0} (A y_{0}t_{f})^2 \left[ 4 + 3\omega \left( 1 - n^2\right) \right] ^2 }{\left( 3 + 2 \omega \right) \left[ 4 - 6 n + 3\omega \left( n - 1\right) ^2\right] }. \end{aligned}$$
(19)

where \(a(t_f)\), \(\phi _f\), and \(t_f\) are constants. In this parameterization, the \(n = 0\) models become identical to those presented in the original BD paper [88]. For \(n \ne 0\), we recover the type A-I solutions discussed in [71], although in a slightly different notation. As shown in [73], the range of \(\omega\) is \(- 2< \omega < -3/2\) \((1< l < 2)\) (within this interval, it is allowed a matter-dominated universe with accelerated expansion). For the radiation-dominated epoch, it is \(\omega = - 3/2\) \((l = 1/2)\). In our analysis, we relax such range being mainly interested to study the modifications of cosmology during the pre-BBN, keeping hence free all parameters of the 5D cosmological model.

3 Gravitational baryogenesis in modified 5D to 4D-cosmology

A suitable scheme mechanism for generating baryon asymmetry during the expansion of the universe has been developed within supergravity theories [89, 90]. The interaction responsible for the CPT violation is given by a coupling between the derivative of the Ricci scalar curvature R and the baryon current \(J^\mu\) [91]

$$\begin{aligned} \frac{1}{M_*^{2}}\int \textrm{d}^4x\sqrt{-g}\,J^\mu \partial _\mu R, \end{aligned}$$
(20)

where \(M_*\) is the cutoff scale characterizing the effective theory (see Ref. [93,94,95,96,97,98,99,100,101,102,103,104,105,106] for further applications).

In order to get interactions that violate the baryon number B in thermal equilibrium (to satisfy the first Sakharov condition), it is required that a net baryon asymmetry should be generated and get frozen in below the decoupling temperatureFootnote 3\(T_D\). From Eq. (20), one gets [91]

$$\begin{aligned} \frac{1}{M_*^2}J^\mu \partial _\mu R=\frac{1}{M_*^2}(n_B-n_{\bar{B}})\dot{R}, \end{aligned}$$
(21)

where

$$\begin{aligned} \mu _B=-\mu _{\bar{B}}=-\frac{\dot{R}}{M_*^2}. \end{aligned}$$
(22)

For relativistic particles, the net baryon number density of matter in the early Universe is given by [92] \(n_B-n_{\bar{B}}=\frac{g_b}{6}\mu _BT^2\), where \(g_b\sim \mathcal {O}(1)\) is the number of intrinsic degrees of freedom of baryons, so that the baryon asymmetry reads

$$\begin{aligned} \eta \equiv \frac{n_B-n_{\bar{B}}}{s}\approx \frac{n_B}{s} \simeq -\frac{15\,g_b}{4\pi ^2g_*}\frac{\dot{R}}{M_*^2T}\bigg |_{T_D}~, \end{aligned}$$
(23)

where \(s=\frac{2\pi ^2g_{*s}}{45}T^3\) is the entropy per unit volume, i.e. entropy density, in the radiation-dominated era, and \(g_{*s}\sim 106\) is the number of degrees of freedom for particles which contribute to the entropy of the Universe [92]. As arises from (23), the parameter \(\eta\) is different from zero provided that \(\dot{R}\ne 0\). In the radiation-dominated era, described by GR, the baryon asymmetry is zero (\(\eta =0\)) because in that case \(\dot{R}=0\). 5D-corrections modify \(\dot{R}\), making \(\eta \ne 0\). To obtain the 5D-corrected derivative of the Ricci scalar \(\dot{R}\), we need first to compute the trace of the Einstein equation \(R=-8\pi G T_g =-8\pi G\,(\rho -3p)\), where \(T_g =\rho -3p\) is the trace of the energy-momentum tensor of matter. Such a term disappears in GR since the radiation equation of state of relativistic particles is given by \(p=\rho /3\). In a modified 5D-cosmology, the total energy-momentum tensor contains both matter and scalar fields, so that the total trace does no vanish owing to the scalar field contribution, as we will see. By using (18), one gets

$$\begin{aligned} R=-6({\dot{H}}+2H^2)=\frac{6\alpha (2\alpha -1)}{t^2}, \quad \alpha =\frac{2[1+\omega (1-n)]}{4+3\omega (1-n^2)}. \end{aligned}$$
(24)

where \(H={\dot{a}}/a\) is the expansion rate of the Universe. The time derivative of (24) gives

$$\begin{aligned} \dot{R}=12\alpha (1-2\alpha )\left( \frac{16 \pi ^2 g_*}{45}\right) ^{3/2} \left( \frac{T_f^2}{M_P}\right) ^3 \left( \frac{T}{T_f}\right) ^{3/\alpha }. \end{aligned}$$
(25)

where \(T_f\) (or \(t_f\)) is the temperature (times) in which the Universe starts to evolve according to GR. To obtain (25), we have assumed entropy conservation so that \(T a= T_0\) (we set \(a_0=1\) and \(T_0\simeq 10^{-4}\)eV). Such a transition occurs at BBN or well before that time (the so-called pre-BBN era). Substituting Eqs. (25) into (23), one obtains the expression for the baryon asymmetry

$$\begin{aligned} \eta =\frac{425 g_b}{\pi ^2 g_*} \left( \frac{16\pi ^2 g_*}{45}\right) ^{3/2} \left( \frac{T_f}{M_P}\right) ^{3(2\alpha -1)/\alpha } \left( \frac{M_P}{M_*}\right) ^2 \left( \frac{T_D}{M_P}\right) ^{(3-\alpha )/\alpha }. \end{aligned}$$
(26)

Figure 1 is reported \(\eta\) in (26) as function of \(\alpha\) for two significant cases of the decoupling temperature, \(T_D\sim T_{GUT}\sim 10^{15}\)GeV and \(T_D \sim 10^2\)GeV. We have fixed \(g_b=2\), \(g_*\sim 106\), \(M_P\sim 10^{19}\)GeV, and the transition temperature (from pre-BBN Universe described by modified 5D-cosmology to the standard cosmology) to \(T_f \sim T_{BBN}\sim 1\)MeV. Let us discuss the results displayed in this picture from high temperature \(T_D \sim T_{GUT} \sim 10^{15}GeV\) to \(T_D \sim 10^2\)GeV (below this temperature, the sphaleron effects are no more operative):

  • \(T_D \sim T_{GUT} \sim 10^{15}GeV\)—From Fig. 1(Left), it follows that the exponential parameter \(\alpha\) turns out to be of the order \(\alpha \sim 0.49\), i.e. it corresponds to a tiny deviation from the standard cosmological evolution of the Universe for which \(\alpha =1/2\) (radiation dominated era).

  • \(T_D \sim 10^2\)GeV—The observed baryon asymmetry follows for \(\alpha \sim 0.161\), Fig. 1(Right). This requires \(\omega <0\). According to Eq. (18), that give \(\alpha\) and \(\omega\) vs l, negative values of \(\omega\) are allowed by the cosmological model under consideration, as shown in Fig. 2.

Fig. 1
figure 1

Left—\(\eta\) vs \(\alpha\) for fixed values \(T_f=T_{BBN}=1\)MeV, \(M_*=10^{15}\)GeV, and the decoupling is assumed to occur at GUT scales \(T_D=10^{14}\)GeV. Right—\(\eta\) vs \(\alpha\) for fixed values \(T_f=T_{BBN}=1\)MeV, \(M_*=10^{15}\)GeV, and the decoupling is assumed to occur at low scales \(T_D=10^{2}\)GeV

Full size image
Fig. 2
figure 2

\(\alpha\) and \(\omega\) vs l, Eq. (18). Here, we fixed the adiabatic index to \(n=0.333\) (it is a tiny deviation from the adiabatic index in the era radiation dominated \(n=1/3\)). As we can see, \(\alpha \simeq 0.16\) for \(l\simeq 0.16\), to which corresponds \(\omega \simeq -1.5\). Notice that \(\omega\) has an asymptotic at \(l\simeq 0.5\) for \(n\sim 1/3\)

Full size image

4 Primordial light element \(\{ {}^4 He, D, Li\}\) in modified 5D-cosmology

In this section, we analyse the effects of the Brans–Dicke induced-matter cosmology on the primordial light element formed in the early stage of the Universe. We consider the scale factor (18) where \(t_f\) \((T_f)\) is interpreted as the instant (temperature) at which the Universe starts to evolve according to GR. We can write down the modified 4D-hypersurface expansion rate \(H={\dot{a}}/a\) in terms of the expansion rate \(H_{GR}\) of GR as

$$\begin{aligned} H(T)\equiv Z(T)\,H_{\text {GR}}(T)\,. \end{aligned}$$
(27)

Here \(H_{GR}=\displaystyle {\sqrt{\frac{8\pi }{3M_{Pl}}\, \rho (T)}}\) is the usual expansion rate of the Universe in GR, with \(\rho =\frac{\pi ^2 g_*}{30}\, T^4\), while the factor Z(T) is defined as

$$\begin{aligned} Z(T) \equiv \eta \left( \frac{T}{T_{f}}\right) ^\nu \,, \quad \eta \equiv 2\alpha \,, \quad \nu \equiv \frac{1-2\alpha }{\alpha }\,. \end{aligned}$$
(28)

For \(\alpha =1/2\), it follows \(\nu =0\), \(\eta =1\), so that \(Z(T)=1\) and GR is recovered. Owing to the modified expansion rate induced by the new sources of higher dimensional within the field equations, our goal is to infer the bounds on the parameter \(\alpha\) that controls scale factor evolution from the primordial abundances of light elements, that is deuterium \({}^{2}H\), helium \({}^{4}He\), and litium \({}^{7}Li\). To this aim, one replaces the Z-factor entering the expressions of the primordial light elements (related to the effective number of neutrinos species, \(Z_\nu = \left[ 1 + \frac{7}{43} (N_{\nu } - 3)\right] ^{1/2}\) [107]) with the Z(T) factor entering (28) [108, 109]. Being interested in deviations from the standard cosmology, hereafter, we shall assume that the number of neutrino generations is \(N_{\nu } = 3\). Following the approach given in [110], we can observe:

  • \(^{4}He\) abundance—The relevant reactions for the BBN processes giving \(^{4}He\) are

    $$\begin{aligned} n + p \rightarrow {}^{2}H + \gamma ; \qquad {}^{2}H + {}^{2}H \rightarrow {}^{3}He + n; \qquad {}^{2}H + {}^{2}H \rightarrow {}^{3}H + p \end{aligned}$$
    (29)

    while the reactions

    $$\begin{aligned} {}^{2}H + {}^{3}H \rightarrow ^{4}He + n \quad \text {and} \quad {}^{2}H + {}^{3}He\rightarrow {}^{4}He + p \end{aligned}$$
    (30)

    produce the helium \({}^{4}He\). The best fit of the primordial \({}^{4}He\) abundance is given by [111, 112]

    $$\begin{aligned} Y_{p} = 0.2485 \pm 0.0006 + 0.0016 \left[ \left( \eta _{ 10} - 6\right) +100\left( Z-1\right) \right] \,. \end{aligned}$$
    (31)

    Here Z is defined in (28), and \(\eta _{10}\) is given by [108, 109]

    $$\begin{aligned} \eta _{10} \equiv 10^{10}\eta _{B}\equiv 10^{10} \frac{n_{B}}{n_{\gamma }}\,, \quad \eta _{10} \simeq 6\,. \end{aligned}$$
    (32)

    The quantity \(\eta _{B}=n_B/n_\gamma\) is the baryon to photon ratio [113]. The value \(Z=1\) refers to the standard BBN results in GR for the \(^{4}He\)-fraction, which means \((Y_{p})|_{GR} = 0.2485 \pm 0.0006\). The observed data relative to the helium \(^{4}He\) and the value \(\eta _{10} = 6\) provide the abundance [114]

    $$\begin{aligned} 0.2561 \pm 0.0108 = 0.2485 \pm 0.0006 + 0.0016 \left[ 100(Z-1) \right] \,. \end{aligned}$$
    (33)

    These results imply the constrain

    $$\begin{aligned} Z = 1.0475 \pm 0.105\,. \end{aligned}$$
    (34)

  • \(^{2}H\) abundance—The reaction producing deuterium \(^{2}H\) is given by \(n+p \rightarrow {}^{2}H +\gamma\). The best fit of deuterium abundance gives [108]

    $$\begin{aligned} y_{D p} = 2.6(1 \pm 0.06) \left( \frac{6}{\eta _{ 10} - 6 (Z-1)}\right) ^{1.6} \end{aligned}$$
    (35)

    The standard result in GR is \(y_{D p} |_{GR} = 2.6 \pm 0.16\), that follows by setting \(Z=1\) and \(\eta _{10} = 6\). From observation with (35), one gets [114]

    $$\begin{aligned} 2.88 \pm 0.22 = 2.6 (1 \pm 0.06) \left( \frac{6}{\eta _{ 10} - 6 (Z-1)}\right) ^{1.6}\,, \end{aligned}$$
    (36)

    which implies

    $$\begin{aligned} Z =1.062 \pm 0.444\,. \end{aligned}$$
    (37)

    Such a constraint partially overlaps with the helium abundance (34).

  • \(^{7}Li\) abundance—The ratio between the \({}^{7}Li\) abundance in GR and the observed is \(\frac{Li|_{GR}}{Li|_{obs}} \in [2.4-4.3]\) [107], the best fit for \(^{7}Li\) abundance is [108]

    $$\begin{aligned} y_{Li} = 4.82 (1 \pm 0.1)\left[ \frac{\eta _{ 10} - 3 (Z-1)}{6} \right] ^{2}\,. \end{aligned}$$
    (38)

    The constraints on lithium abundance \(y_{Li} = 1.6 \pm 0.3\) [114] imply

    $$\begin{aligned} Z = 1.960025 \pm 0.076675\,. \end{aligned}$$
    (39)

    Notice that this value does not overlap with the constraints on \({}^2 H\) and \({}^4He\) abundance given in Eqs. (37) and (34). Moreover, it must be mentioned that the 5D BD cosmological model does not allow to solve the the lithium problem [107], related to the fact that \(\eta _{10}\) given in (32) allows to successfully fit the abundances of D and \({}^4He\), but does not fit the observations of \(^{7}Li\).

Fig. 3
figure 3

Left—\(Z_{{}^4 He}\) vs \(\alpha\). The range (34) is reported. The baryon parameter is fixed to \(\eta _{10}=6\) \(T_f= 0.1\)MeV, and we have taken the temperature \(T=\{1, 10\}\)MeV. Right—\(Z_{Li}\) vs \(\alpha\). The experimental bounds (39) are reported. The other values are the same of Left panel

Full size image
Fig. 4
figure 4

Left—\(Z_{{}^4 He}\) vs T. The experimental bounds (34) are reported. Here \(\alpha =0.16\), the baryon parameter is fixed to \(\eta _{10}=6\), and \(T_f= 0.1\) MeV. Right—\(Z_{Li}\) vs T. The experimental bounds (39) are reported. The other values are the same of Left panel

Full size image

In Fig. 3—Left, we plot the factor Z(T) given in (28) taking into account the constraints (34). We assume \(\eta _{10}=6\) and \(T_f=0.1\)MeV. The plots show the behaviour of the factor \(Z_{He}\) for typical BBN temperatures, \(T=1\)MeV and \(T=10\)MeV. One gets that \(Z_{He}\ne 1\) for \(\alpha\) ranging in the intervals

$$\begin{aligned} 0.48\lesssim \alpha \lesssim 0.51{} & {} \quad \text {for} \,\, T=1\text {MeV} , \nonumber \\ 0.495\lesssim \alpha \lesssim 0.501{} & {} \quad \text {for} \,\, T=10\text {MeV} , \end{aligned}$$
(40)

A similar result is inferred for the deuterium using the constraints on Z descending from data on this element Eq. (37).

In the case of lithium, the plot provided in Fig. 3—Right, using the constraints on Z given by Eq. (39), indicates that \(Z_{Li}\ne 1\) for \(\alpha\) in the range

$$\begin{aligned} 0.42\lesssim \alpha \lesssim 0.45{} & {} \quad \text {for} \,\, T=1\text {MeV} , \nonumber \\ 0.46\lesssim \alpha \lesssim 0.47{} & {} \quad \text {for} \,\, T=10\text {MeV} . \end{aligned}$$
(41)

The results obtained in Eqs. (40) and (41) do not overlap. However, the difference between the different ranges is very tiny, suggesting that the lithium problem could be ameliorated in the modified cosmology framework.

We finally discuss the case in which the value of the parameter \(\alpha\) is fixed to \(\alpha =0.16\), which is relevant for successful baryogenesis at a low-energy scale. Figure 4 is reported the behaviour of \(Z_{He}\) and \(Z_{Li}\) vs T, for \(T_f=0.1\)MeV. As a result, \(Z(T)\ne 1\) in temperature range of the order \(T\in [0.130, 0.135]\)MeV for He and \(T\in [0.152, 0.156]\)MeV for Li.

5 Conclusions

In this paper, we have considered a higher dimensions cosmological model, in particular 5D BD theory. In this context, it is possible to show that the BD field equations in 5D are equivalent to those of BD in 4D with some new ingredients in the sources side deriving from higher dimensional counter-terms. In the framework of FRW Universe, field equations lead to different classes of solutions. In particular, it is possible to obtain a power-law solution \(a(t)\sim t^\alpha\) of the scale factor under specific conditions. Such a power-law solution has been used to investigate the baryon asymmetry in the Universe induced by a gravitational mechanism. The effective mechanism is the coupling of baryon currents to the Ricci scalar, which induces the CPT symmetry violation since the Ricci curvature depends on time also in the radiation regime. This class of solutions has been also studied for the formation of primordial light elements obtaining a constraint on the parameter \(\alpha\) that controls the scale factor evolution.

The 5D-modified model opens interesting cosmological scenarios with relevant consequences in different contexts, such as, for example, the thermal dark matter (DM) freeze-out mechanism, which allows deriving a bound on \(\alpha\) from the observed DM relic abundance (more precisely, the bound on \(\alpha\) can be inferred by using the dark matter (DM) annihilation cross-section which enters the cold DM relic abundance \(\Omega _{cdm}\)). By comparing the factor Z(T) in (28) with that one of Ref. [115], \(Z(T) \sim T^{\frac{2}{n}-2}\), one gets \(\alpha =n/2\). For DM composed only by weakly interacting massive particles (WIMPs), the DM relic density in modified cosmology reads [115]

$$\begin{aligned} \Omega _{cdm}h^{2} \simeq 10^{9} \frac{(\bar{l} + 1) x_{f}^{(\bar{l} + 1)} \text {GeV}^{-1}}{(h_{*}/g_{*s}^{1/2}) M_{P}\bar{\sigma }}\,, \end{aligned}$$
(42)

where \(\bar{l} = l + \left( 2\alpha -1 \right)\) (notice that l here is not related to the exponent appearing in (14) but it refers to angular momentum), \(\bar{\sigma }\) is the WIMP cross-section, \(h_*\) is the number of relativistic degrees of freedom for entropy density, and \(x_{f}\equiv m/T_F\) (\(T_F\) is the freeze-out temperature) [115, 116]

$$\begin{aligned} x_{f} = \ln [0.038 (\bar{l} + 1) (g / g_{*}^{1/2}) M_{p} m \bar{\sigma }] - (\bar{l} + 1) \ln [\ln [0.038 (\bar{l} + 1) (g / g_{*}^{1/2}) M_{p} m \bar{\sigma }]]\,, \end{aligned}$$
(43)

with \(g=2\) the spin polarizations of the dark matter particle and m the mass of WIMPs particles. HereFootnote 4\(\bar{l} = l\) (\(\alpha = 1/2\)) for GR (\(l=0, 1\) corresponds to s-wave and p-wave polarizations). Since, cosmological data on cold dark matter density give \(\Omega _{cdm}h^{2}=0.1198 \pm 0.0012\) [118, 119], one infers \(|1-2\alpha | \lesssim 1.6 \times 10^{-4}\). Therefore, the DM relic density allows a tiny deviation from the standard cosmological model as a counter-effect of 5D Brans–Dicke cosmology. Interestingly, this bound is compatible with the allowed values for \(\alpha\) obtained from He during the light element formation and partially from gravitational baryogenesis at GUT scales (for higher decoupling temperatures the parameter \(\alpha\) approaches to 1/2).

In conclusion, modified 5D cosmological models offer interesting possibilities for studying their consequences in different scenarios. Since the strong analogy between Brans–Dicke models and higher-order theories of gravity, for example, the present analysis could represent a straightforward check also for such kinds of models. A general analysis from this point of view will be faced elsewhere.