Noether symmetry approach in non-minimal derivative coupling gravity

Abstract

In this work, we examine solutions of the system of equations obtained by applying the Noether gauge symmetry (NGS) and its conserved quantity for the standard general relativity (GR) and the non-minimal derivative coupling (NMDC) cosmological model. We discover two salient features of the solutions. The first one is $a(t)\propto t^{1/3}$ for a kinetic-dominant phase which may emerge before inflationary period at very early time for GR case. The second one is a new form of scalar field $\phi (t)$ govern by the exponential cosmological solution for NMDC case, $\phi (t)=(c_{1}+c_{2}t)e^{-\lambda t}+c_{3}$.

1 Introduction

Astrophysical observations including Type Ia Supernovae [1, 2], cosmic microwave background (CMB) radiation [3,4,5,6,7,8,9], large scale structure [10], baryon acoustic oscillations (BAO) [11] as well as weak lensing [12] provide a strong evidence that the expansion of the universe is presently accelerating. In spite of its successes, the so-called Lambda cold dark matter ($\Lambda$CDM) [13] is plagued by the cosmological problem [14] and the coincident problem [15]. The phase of late-time cosmic acceleration receives much attention. However, the introduction of the so-called “dark energy (DE)” in the context of conventional general relativity is one of the promising explanations. Additionally, another possible scenario is to engineer Einstein gravity on the large-scale methodology. There were some reviewed articles published so far regarding the mentioned issues, see for example [16,17,18,19] and references therein. However, very little is known about the DE sector of the universe and it poses one of the unsolved problems in physics.

Alternative paradigms by engineering the Einstein field equations either in the geometric part or in the stress-energy tensor are widely accepted to explain effects of dark ingredients [20]. The f(R) theories of gravity serves as one of the simplest modifications to the standard general relativity. Here the Lagrangian density of f is an arbitrary function of the scalar curvature R [21, 22]. It is worth noting that there were rigorous reviews on f(R) theories [23, 24] as well as on Born–Infeld inspired modifications of gravity [25]. In Ref. [26], the authors investigated the cosmological implications of the modified theories of gravity on inflation, bounce and late-time evolution. Apart from these modified theories of gravity, theories of non-minimal derivative coupling to gravity attract much attention from theoretical and phenomenological points of view, see, e.g., [27,28,29,30,31,32,33,34,35,36,37,38]. More specifically, their applications on inflation and its consequences were proposed by a number of authors [39,40,41,42,43,44,45,46,47].

The important role of the Noether symmetry in cosmology has received increasing attention within decades in order to select the viable models [48]. Conserved quantities of the system, as well as unknown functions, can be determined with the help of the Noether symmetry approach. More specifically, by using the Noether symmetry, we can obtain the exact solutions. The Noether symmetry approach has been applied to study various cosmological scenarios so far including nonlocal f(T) gravity [49,50,51], viable mimetic f(R) and f(R, T) theories [52], f(R) cosmology [53], the cosmological alpha-attractors [54],and $f({\mathcal {G}})$ theory [55]. Moreover, the exact solutions for potential functions, scalar field and the scale factors in the Bianchi models have been investigated in Refs. [56,57,58] and the solutions of the field equations of f(R) gravity are investigated in static cylindrically symmetric space-time using the Noether symmetry technique [59].

The second kind of Noether symmetry approach for cosmological studies in the literature is the so-called Noether gauge symmetry (NGS) approach [60,61,62]. It is a generalization of the conventional one. Very recently, the authors of Ref. [63] have discussed the NGS approach for the Eddington-inspired Born–Infeld theory. In the present work, we study a formal framework of the non-minimal derivative coupling (NMDC) gravity scenario through the NGS approach and present a detailed calculation of the point-like Lagrangian. The point-like Lagrangian of the Einstein-Hilbert action including the non-minimal derivative coupling (NMDC) sector are examined with spatially flat FLRW spacetime in which matter in such a universe only has a scalar field and a matter field. The latter model is expected to quantify to what extend the field kinetic term affects the evolution of the universe.

This paper is organized as follows: We will start by making a short recap of a formal framework of NMDC gravity and derive the point-like Lagrangian for underlying theory in Sect. 2. In Sect. 3, we study a Hessian matrix and quantify the Euler–Lagrange equations and Hamiltonian equations of the Einstein (GR) and the NMDC universes. In Sect. 4, the NGS approach for the GR and NMDC is discussed. We discuss exact cosmological solutions of both theories with the help of the Noether symmetries of point-like Lagrangian. Finally, we conclude our findings in the last section.

2 Non-minimal-derivative coupling gravity

The non-minimal derivative coupling (NMDC) gravity model is a special case of the fifth term of Horndeski Lagrangian density that is given as follows:

$$\begin{aligned} {\mathcal {L}}_{5}=G_{5}(\phi ,X)G^{\mu \nu }\partial _{\mu }\partial _{\nu }\phi , \end{aligned}$$

(2.1)

where $X=-\frac{1}{2}\partial _{\mu }\phi \partial ^{\mu }\phi$. If we set $G_{5}(\phi ,X)=-\phi /(M^{2})$ where M is the a new energy scale in the theory. Performing an integration by parts, we obtain

(2.2)

The appearance of minus sign in front of $\frac{1}{M^{2}}G^{\mu \nu }$ is due to the avoidance of ghosts in the scalar field sector, i.e., ${\dot{\phi }}^{2}+2V(\phi ) >{\frac{1}{M^{2}}}G_{00}g^{00}g^{00}{\dot{\phi }}^{2}=\frac{3}{M^{2}}(\frac{{\dot{a}}}{a})^{2}{\dot{\phi }}^{2}.$ Hence, the potential term dominates over the NMDC term [42]. The NMDC Lagrangian was first proposed in [27] and the action is given by

$$\begin{aligned}&S_{\mathrm{NMDC}}(g)\nonumber \\&\quad =\int d^4x \sqrt{-g} \bigg [{R}-(\epsilon g_{\mu \nu }+\kappa G_{\mu \nu })\phi ^{,\mu }\phi ^{,\nu }-2V(\phi ) \bigg ]\nonumber \\&\qquad +S_{m}(g_{\mu \nu },\Psi ), \end{aligned}$$

(2.3)

where $\kappa \equiv M^{-2}$ is a NMDC free parameter that has dimension of $[M^{-2}_{P}]$,and $S_{m}(g_{\mu \nu },\Psi )$ denotes the matter field action. It is worth mentioning here that the NMDC-point-like Lagrangian is in accordance with the symmetry-class J in Table I [74]. However, in our paper, it is not necessary to set $V(\phi )$ to be a constant in order to just conform to $g_{2}(X)$, while $V(\phi )$ is set to $V(\phi )=\Lambda ={\mathrm{const}}$ given just below Eq. (64) of Ref. [74]. The spatially flat FLRW metric can be written as

$$\begin{aligned} ds^{2}_{g}= & {} g_{\mu \nu }dx^{\mu }dx^{\nu }= -dt^{2}+a(t)^{2}d{\vec {x}}^{2}, \end{aligned}$$

(2.4)

where $\upsilon _{0}\equiv d^{3}x$ is the spatial volume after a proper compactification for spatial flat section. $\epsilon =+1$ and $\epsilon =-1$ denote an ordinary scalar field and phantom scalar field, respectively. The point-like Lagrangian can be extracted from Eq. (2.3) to yield

$$\begin{aligned} {\mathcal {L}}_{\mathrm{NMDC}}(a,\phi ,{\dot{a}},{\dot{\phi }})= & {} -6a{\dot{a}}^2+\bigg ({{\epsilon }a^3{\dot{\phi }}^2}-3\kappa a{\dot{a}}^2 {\dot{\phi }}^2 \bigg ) \nonumber \\&-2a^3 V(\phi )-2\rho _{m}(a)a^3. \end{aligned}$$

(2.5)

It is easy to see that the number of configuration space (n) (or the minisuperspace) is equal to 2 because of the appearance of variables $\{a(t),\phi (t)\}$ in ${\mathcal {L}}_{\mathrm{NMDC}}$.

3 Hessian matrix, EL equations and NMDC universe

The configuration space variables and their time derivative of both gravity models are $q^{i}= \{a,\phi \}$ and ${\dot{q}}^{i}=\frac{dq_{i}}{dt} =\{{\dot{a}},{\dot{\phi }}\}.$ The Hessian matrix for GR by using the point-like Lagrangian, ${\mathcal {L}}_{GR}=-6a{\dot{a}}^{2}+\epsilon a^{3}{\dot{\phi }}^{2}-2a^{3}V(\phi )-2\rho _{m}(a)a^{3},$ can be expressed as

$$\begin{aligned}{}[W_{ij}]_{\mathrm{GR}}=\left[ \begin{array}{cc} \frac{\partial ^{2}L}{\partial {\dot{a}}^{2}} &{}\quad \frac{\partial ^{2}L}{\partial {\dot{a}}\partial {\dot{\phi }}} \\ \frac{\partial ^{2}L}{\partial {\dot{\phi }}\partial {\dot{a}}} &{}\quad \frac{\partial ^{2}L}{\partial {\dot{\phi }}^{2}} \end{array} \right] = \left[ \begin{array}{cc} -12a &{} 0 \\ 0 &{} 2\epsilon a^{3} \\ \end{array} \right] . \end{aligned}$$

(3.1)

The determinant of the Hessian matrix of the GR Lagrangian is ${\mathrm{det}}[W_{ij,GR}] =-24\epsilon a^{4}\ne 0.$ and the Hessian matrix for NMDC can be expressed as

$$\begin{aligned}{}[W_{ij}]_{\mathrm{NMDC}}= & {} \left[ \begin{array}{cc} \frac{\partial ^{2}{L}}{\partial {\dot{a}}^{2}} &{}\quad \frac{\partial ^{2}{L}}{\partial {\dot{a}}\partial {\dot{\phi }}} \\ \frac{\partial ^{2}{L}}{\partial {\dot{\phi }}\partial {\dot{a}}} &{}\quad \frac{\partial ^{2}{L}}{\partial {\dot{\phi }}^{2}} \end{array} \right] \nonumber \\= & {} \left[ \begin{array}{cc} -12a-6\kappa a{\dot{\phi }}^2 &{}\quad -6\kappa a{\dot{a}}{\dot{\phi }} \\ -6\kappa a{\dot{a}}{\dot{\phi }} &{}\quad 2\epsilon {a^3}-6\kappa a{\dot{a}}^{2} \\ \end{array} \right] . \end{aligned}$$

(3.2)

The determinant of the Hessian matrix of NMDC Lagrangian are ${\mathrm{det}}[W_{ij,NMDC}] =-24\epsilon a^{4}+72\kappa a^{2}{\dot{a}}^{2}-12\kappa \epsilon a^{4}{\dot{\phi }}^{2}\ne 0.$ Without the contribution of NMDC free parameter ($\kappa$), this can be clearly reduced to the parameters derived in GR case. Mathematically, the fact that the determinant of the Hessian matrix is not equal to zero is called a regular or a non-degenerate Lagrangian. It is important to note that a key concept of a gauge field theory is the general solution of the equations of motion that contains arbitrary functions of time and the canonical variables are not all independent but related to each other by the constraint equations [64]. The Hamiltonian constraint equation can be straightforwardly derived from the canonical momenta via the Legendre transformation and Lagrangian as follows:

$$\begin{aligned} {\mathcal {H}}_{\mathrm{GR}}= & {} \frac{\partial {{\mathcal {L}}_{\mathrm{GR}}}}{\partial \dot{q_i}}\dot{q_i}-{\mathcal {L}}_{\mathrm{GR}} =\frac{\partial {\mathcal {L}}_{\mathrm{GR}}}{\partial {\dot{a}}}{\dot{a}}+\frac{\partial {\mathcal {L}}_{\mathrm{GR}}}{\partial {\dot{\phi }}}{\dot{\phi }}-{{\mathcal {L}}_{\mathrm{GR}}},\nonumber \\= & {} p_{a}{\dot{a}}+p_{\phi }{\dot{\phi }}-{{\mathcal {L}}_{\mathrm{GR}}},\nonumber \\= & {} \Big (-{12a{\dot{a}}} \Big ){\dot{a}}-( 3\epsilon \kappa a {\dot{a}}^2 {\dot{\phi }}){\dot{\phi }}+2\epsilon a^{3}{\dot{\phi }}-{{\mathcal {L}}_{\mathrm{GR}}}=0, \nonumber \\= & {} -{6a{\dot{a}}^2} +{\epsilon a^3 {\dot{\phi }}^2 } +2a^3V(\phi ) +2\rho _m(a)a^3=0.\nonumber \\ \end{aligned}$$

(3.3)

The energy function that is a constant of motions is given by $E_{{\mathcal {L}}} \equiv \frac{\partial {{\mathcal {L}}}}{\partial {\dot{q}}_{i}}{\dot{q}}_{i}-{\mathcal {L}}=p_{i}{\dot{q}}_{i}-\mathcal{{L}}\equiv {{\mathcal {H}}}$ where $p_{i}$ is the canonical momenta. This condition can be rearranged to give the Friedmann equation in the GR case as follows:

$$\begin{aligned} 3H_{\mathrm{GR}}^{2}=\rho _{m}(a)+\frac{\epsilon }{2}{\dot{\phi }}^{2}+V(\phi ). \end{aligned}$$

(3.4)

The Hamiltonian constraint equation for the NMDC model can be found similarly as follows:

$$\begin{aligned} {\mathcal {H}}_{\mathrm{NMDC}}= & {} \frac{\partial {{\mathcal {L}}_{\mathrm{NMDC}}}}{\partial \dot{q_i}}\dot{q_i}-{\mathcal {L}}_{\mathrm{NMDC}} =\frac{\partial {\mathcal {L}}_{\mathrm{NMDC}}}{\partial {\dot{a}}}{\dot{a}}\nonumber \\&+\frac{\partial {\mathcal {L}}_{\mathrm{NMDC}}}{\partial {\dot{\phi }}}{\dot{\phi }}-{{\mathcal {L}}_{\mathrm{NMDC}}},\nonumber \\= & {} p_{a}{\dot{a}}+p_{\phi }{\dot{\phi }}-{{\mathcal {L}}_{\mathrm{NMDC}}},\nonumber \\= & {} \Big (-{12a{\dot{a}}}-6\epsilon \kappa a{\dot{a}}{\dot{\phi }}^2 \Big ){\dot{a}}-( 3\epsilon \kappa a {\dot{a}}^2 {\dot{\phi }}){\dot{\phi }}+2\epsilon a^{3}{\dot{\phi }}\nonumber \\&-{{\mathcal {L}}_{\mathrm{NMDC}}}=0, \nonumber \\= & {} -{6a{\dot{a}}^2} -{9\epsilon \kappa a {\dot{a}}^2 {\dot{\phi }}^2} +{\epsilon a^3 {\dot{\phi }}^2 } +2a^3V(\phi )\nonumber \\&+2\rho _m(a)a^3=0. \end{aligned}$$

(3.5)

This condition can be rearranged to gives the modified Friedmann equation [65] as

$$\begin{aligned} 3H^{2}=\rho _{m}(a)+\frac{\epsilon }{2}{\dot{\phi }}^{2}+V(\phi )-\frac{9}{2}\epsilon \kappa H^{2}{\dot{\phi }}^{2}. \end{aligned}$$

(3.6)

It is easy to notice that it brings up the Friedmann equation for GR when $\kappa =0$. The total NMDC Hamiltonian $({\mathcal {H}}_{\mathrm{NMDC}})$ can be used to evaluate an evolution invoking the Hamiltonian equations of motion as follows:

$$\begin{aligned} p_{a}= & {} \frac{\partial {\mathcal {L}}_{\mathrm{NMDC}}}{\partial {{\dot{a}}}}={\dot{a}}\Big [-12a-6\epsilon \kappa a{\dot{\phi }}^{2} \Big ], \end{aligned}$$

(3.7)

$$\begin{aligned} p_{\phi }= & {} \frac{\partial {\mathcal {L}}_{\mathrm{NMDC}}}{\partial {{\dot{\phi }}}}={\dot{\phi }}\Big [ 2\epsilon a^{3}-6\epsilon \kappa a {\dot{a}}^{2} \Big ]\end{aligned}$$

(3.8)

$$\begin{aligned} {\dot{a}}= & {} \frac{\partial {\mathcal {H}}_{\mathrm{NMDC}}}{\partial p_{a}}=\frac{-p_{a}}{\Big [12a+6\epsilon \kappa a{\dot{\phi }}^{2}\Big ]}\end{aligned}$$

(3.9)

$$\begin{aligned} {\dot{\phi }}= & {} \frac{\partial {\mathcal {H}}_{\mathrm{NMDC}}}{\partial p_{\phi }}=\frac{p_{\phi }}{\Big [ 2\epsilon a^{3}-6\epsilon \kappa a{\dot{a}}^{2}\Big ]}\end{aligned}$$

(3.10)

$$\begin{aligned} {\dot{p}}_{a}= & {} -\frac{\partial {\mathcal {H}}_{\mathrm{NMDC}}}{\partial a}=6a^{2}+9\epsilon \kappa {\dot{a}}^{2}{\dot{\phi }}^{2}-3\epsilon a^{2}{\dot{\phi }}^{2}\nonumber \\&-6a^{2}V(\phi )-2\rho '_{m}(a)a^{3}-6\rho _{m}(a)a^{2}\end{aligned}$$

(3.11)

$$\begin{aligned} {\dot{p}}_{\phi }= & {} -\frac{\partial {\mathcal {H}}_{\mathrm{NMDC}}}{\partial \phi }=-2a^{3}V'(\phi ). \end{aligned}$$

(3.12)

In order to obtain the dynamic solutions, we have to calculate the Euler–Lagrange equations for a(t) and $\phi (t)$ which gives the same results as Eqs. (2)–(4) of Ref. [65] as shown below:

$$\begin{aligned}&6{\dot{a}}^{2}-3\epsilon a^{2}{\dot{\phi }}^{2}+3\kappa {\dot{a}}^{2}{{\dot{\phi }}^{2}}+6a^{2}V(\phi )+2\rho '_{m}(a)a^{3}\nonumber \\&\qquad +6\rho _{m}(a)a^{2}-12{\dot{a}}^{2}-12a\ddot{a}-6\kappa {\dot{a}}^{2}{\dot{\phi }}^{2}\nonumber \\&\qquad -6\kappa a\ddot{a}{\dot{\phi }}^{2}- 12\kappa a{\dot{a}}{\dot{\phi }}\ddot{\phi }=0. \end{aligned}$$

(3.13)

Rearranging, this shows that

$$\begin{aligned} 3H^{2}+2{\dot{H}}= & {} -\frac{{\dot{\phi }}^{2}}{2} \Big [\epsilon +\kappa (2{\dot{H}}+3H^{2}+4H\frac{\ddot{\phi }}{{\dot{\phi }}}) \Big ] +V(\phi )\nonumber \\&+\frac{\rho '_{m}(a)a}{3}+\rho _{m},\nonumber \\= & {} -\frac{{\dot{\phi }}^{2}}{2} \Big [\epsilon +\kappa (2{\dot{H}}+3H^{2}+4H\frac{\ddot{\phi }}{{\dot{\phi }}}) \Big ] +V(\phi )-p_{m},\nonumber \\ \end{aligned}$$

(3.14)

where the fluid equation [66, 67], $\rho '(a)\equiv d\rho _{m}/da=-3(\rho _{m}+p_{m})/a,$ has been used to get the last line of Eq. (3.13).

The modified Klien–Gordon equation for NMDC is directly calculated from the Euler–Lagrange equation for scalar field. Thus this gives

$$\begin{aligned}&2a^{3}V_{\phi }+6\epsilon a^{2}{\dot{a}}{\dot{\phi }}+2\epsilon a^{3}\ddot{\phi }-6\kappa {\dot{a}}^{3}{\dot{\phi }}-12\kappa a{\dot{a}}\ddot{a}{\dot{\phi }}\nonumber \\&\quad -6\kappa a{\dot{a}}^{2}\ddot{\phi }=0. \end{aligned}$$

(3.15)

Having used the relation $\frac{\ddot{a}}{a}={\dot{H}}+H^{2}$, it can be shown that

$$\begin{aligned} \epsilon (\ddot{\phi }+3H{\dot{\phi }})-3\kappa \Big ( H^{2}\ddot{\phi }+2H{\dot{H}}{\dot{\phi }}+3H^{3}{\dot{\phi }} \Big )=-V_{\phi }.\nonumber \\ \end{aligned}$$

(3.16)

4 Noether gauge symmetries

In this section, we employ the Noether gauge symmetries to figure out exact solutions of the systems both in the standard GR cosmology and in the NMDC universe.

4.1 Standard GR cosmology

The Lagrangian for GR is written as

$$\begin{aligned} {\mathcal {L}}_{GR}=\sqrt{-g}{R}+{\mathcal {L}}_{\phi }+{\mathcal {L}}_{m}, \end{aligned}$$

(4.1)

where the second term is the scalar field Lagrangian and the last term is the matter field one. We can derive the point-like Lagrangian of GR to obtain

$$\begin{aligned} {\mathcal {L}}_{GR}= {-6a{\dot{a}}^2}+\epsilon a^{3}{\dot{\phi }}^{2}-2a^{3}V(\phi )-2a^{3}\rho _{m}. \end{aligned}$$

(4.2)

The approach of Noether gauge symmetry can be applied to Eq. (2.5) aiming to specify cosmological functions of the standard GR and NMDC gravity. A vector field in this approach can be written as

$$\begin{aligned} \mathrm {X}_{\mathrm{NGS}}=\tau \frac{\partial }{\partial t}+\alpha \frac{\partial }{\partial a}+\varphi \frac{\partial }{\partial \phi }. \end{aligned}$$

(4.3)

The first prolongation of $N_{\mathrm{NGS}}$ reads

$$\begin{aligned} \mathrm {X}^{[1]}_{\mathrm{NGS}}= & {} \mathrm {X}_{\mathrm{NGS}}+{\dot{\alpha }}\frac{\partial }{\partial {\dot{a}}}+{\dot{\varphi }}\frac{\partial }{\partial {\dot{\phi }}}, \end{aligned}$$

(4.4)

where the undetermined parameter $\tau$ is a function of $\{t,a,\phi \}$. The time derivatives for $\alpha (t,a,\phi )$ and $\varphi (t,a,\phi )$ are defined as

$$\begin{aligned} {\dot{\alpha }}(t,a,\phi )= & {} \mathrm {D}_{t}\alpha -{\dot{a}}\mathrm {D}_{t}\tau ,\nonumber \\ {\dot{\varphi }}(t,\phi )= & {} \mathrm {D}_{t}\varphi -{\dot{\phi }}\mathrm {D}_{t}\tau . \end{aligned}$$

(4.5)

Here $\mathrm {D}_t$ is the operator of a total differentiation with respect to t, i.e.

$$\begin{aligned} \mathrm {D}_t =\frac{\partial }{\partial t}+{\dot{a}}\frac{\partial }{\partial a} +{\dot{\phi }}\frac{\partial }{\partial \phi }. \end{aligned}$$

(4.6)

The vector field $\mathrm {X}_{\mathrm{NGS}}$ is a NGS of a Lagrangian ${{\mathcal {L}}}(t,a,\phi ,{\dot{a}},{\dot{\phi }})$ if there exists a gauge function $\mathrm {B}(t, a, \phi )$ which obeys the Rund–Trautmann identity, see [68,69,70] for explicit derivation

$$\begin{aligned} \mathrm {X}^{[1]}_{NGS}{{\mathcal {L}}}+{{\mathcal {L}}}\mathrm {D}_t\tau =\mathrm {D}_t \mathrm {B}. \end{aligned}$$

(4.7)

For NSG without gauge term, i.e. $\mathrm {B}=0$, it requires that $\tau =0$. Therefore Eq. (4.7) is reduced to that is the condition for Noether symmetry [62].

The Noether gauge condition yields

$$\begin{aligned} X^{[1]}_{NGS}{\mathcal {L}}_{GR}+{\mathcal {L}}_{GR} D_{t}\tau =D_{t}B \end{aligned}$$

(4.8)

Using the Noether gauge condition to ${\mathcal {L}}_{GR}$, we have 24 terms which can be expressed as follows:

$$\begin{aligned} D_tB= & {} -6a^2V(\phi )\alpha -6a^2\alpha \rho _m(a) -{6\alpha {\dot{a}}^2} -2a^3\varphi V'(\phi ) \nonumber \\&+{3\epsilon a^2\alpha {\dot{\phi }}^2} -2a^3\alpha \rho '_m(a) \nonumber \\&-{12a{\dot{a}}\alpha _t} -2a^3V(\phi )\tau _t -2a^3\rho _m(a)\tau _t +{6a{\dot{a}}^2\tau _t} \nonumber \\&-{\epsilon a^3{\dot{\phi }}^2\tau _t} +{2\epsilon a^3{\dot{\phi }}\varphi _t} -{12a{\dot{a}}{\dot{\phi }}\alpha _\phi }\nonumber \\&-2a^3V(\phi ){\dot{\phi }}\tau _\phi -2a^3\rho _m(a){\dot{\phi }}\tau _\phi +{6a{\dot{a}}^2{\dot{\phi }}\tau _\phi } \nonumber \\&-{\epsilon a^3{\dot{\phi }}^3\tau _\phi } +{2\epsilon a^3{\dot{\phi }}^2\varphi _\phi }-12a{\dot{a}}^{2}\alpha _{a}\nonumber \\&-2a^{3}V(\phi ){\dot{a}}\tau _{a}-2a^{3}\rho _{m}(a){\dot{a}}\tau _{a}+6a{\dot{a}}^{3}\tau _{a}\nonumber \\&-\epsilon a^{3}{\dot{a}}{\dot{\phi }}^{2}\tau _{a}+2\epsilon a^{3}{\dot{a}}{\dot{\phi }}\varphi _{a} \end{aligned}$$

(4.9)

After separation of monomials and polynomials in term of ${\dot{a}}^2, {\dot{\phi }}^2, {\dot{a}}, {\dot{\phi }}, {\dot{a}}{\dot{\phi }}, {\dot{a}}^2{\dot{\phi }},{\dot{\phi }}^3, {\dot{a}}^3$ and ${\dot{a}}{\dot{\phi }}^2$, the constraints equations and the PDEs can be given as

$$\begin{aligned}&\varphi _a=\alpha _\phi =\tau _\phi =\tau _a=0 \end{aligned}$$

(4.10)

$$\begin{aligned}&\ \alpha +2a\alpha _a-a\tau _t=0 \end{aligned}$$

(4.11)

$$\begin{aligned}&3\alpha +2a\varphi _\phi -a\tau _t=0 \end{aligned}$$

(4.12)

$$\begin{aligned}&6\alpha _{\phi }=\epsilon a^{2}\varphi _{a}=0 \end{aligned}$$

(4.13)

$$\begin{aligned}&B_a=-12a\alpha _{t} \end{aligned}$$

(4.14)

$$\begin{aligned}&B_\phi =2\epsilon a^3\varphi _t \end{aligned}$$

(4.15)

$$\begin{aligned}&B_t=-\alpha \big [ 6a^2V(\phi )+6a^2\rho _m(a) +2a^3\rho '_m(a) \Big ] -2\varphi a^3 V'(\phi )\nonumber \\&\qquad \qquad -\tau _t \Big [ 2a^3V(\phi )+2a^3\rho _m(a) \Big ] . \end{aligned}$$

(4.16)

Table 1 We show possible parameters of the system $\alpha ,\,\varphi$ and $\tau$ and their relations

Because there are no additional conditions, we are interest in examining only for a linear form of solutions. Using the fact that $\alpha _{a}= c_{1}$ and $\varphi _{\phi } = c_{2}=0$, this gives $\tau _{t}=3c_{1}$, the boundary term partly derived from Eqs. (4.14) and (4.15) is expressed as

$$\begin{aligned} B_{(a,\phi )}=-6c_{1}a^{2}{\dot{a}} \end{aligned}$$

(4.17)

The constant of motion for the GR case is given by

$$\begin{aligned} \Sigma _{0,{\mathrm{GR}}}= & {} \alpha \frac{\partial {\mathcal {L}}}{\partial {\dot{a}}}+\varphi \frac{\partial {\mathcal {L}}}{\partial {\dot{\phi }}},\nonumber \\= & {} \alpha ( -12a{\dot{a}} )+\varphi (2\epsilon a^{3}{\dot{\phi }}).\nonumber \\= & {} -12c_{1}a^{2}{\dot{a}} \end{aligned}$$

(4.18)

A first integral or a Noether integral of the system or a conserved quantity associated with X can be derived using

$$\begin{aligned} I(t,q^{i},{\dot{q}}^{i})= & {} -\tau (t,q^{i})\Big ( {\dot{q}}^{i}\frac{\partial {\mathcal {L}}}{\partial {\dot{q}}^{i}}-{\mathcal {L}}\Big )+\eta ^{i}\frac{\partial {\mathcal {L}}}{\partial {\dot{q}}^{i}}\nonumber \\&-B(t,q^{i})\,. \end{aligned}$$

(4.19)

We note that the gauge function $B(t,q^{i})$ in our paper is indeed a function $f(t,q^{i})$ of Ref. [74], However, in our work, it is not a constant, while the authors of [74] have assumed that $f(t,a,\phi )=f_{1}={\mathrm{const.}}$ . Additionally, we will consider below the two forms of the function $B(t,q^{i})$: a linear form, $B_{1}(t)_{\mathrm{NMDC}}=c_{4}t+c_{5}$ and an exponential decay, $B_{2}(t)_{\mathrm{NMDC}}=c_{4}e^{-\lambda t}+c_{5}$. It is easy to see that when setting $\tau =0$ or $\Big ( {\dot{q}}^{i}\frac{\partial {\mathcal {L}}}{\partial {\dot{q}}^{i}}-{\mathcal {L}}\Big )=0$

$$\begin{aligned} I(t,q^{i},{\dot{q}}^{i})= & {} \eta ^{i}\frac{\partial {\mathcal {L}}}{\partial {\dot{q}}^{i}}-B(t,q^{i}),\nonumber \\= & {} \Sigma _{0}-B(t,q^{i}). \end{aligned}$$

(4.20)

where in the GR case $\tau (t)=3c_{1}t+c_{3}$ and $\eta ^{i}=\{\alpha (a)=c_{1}a,\varphi (\phi )=c_{2}\phi =0\}.$ The Lagrangian-related Noether vector X for GR is

$$\begin{aligned} X=\xi (t,q^{k})\frac{\partial \mathcal {}}{\partial t}+\eta ^{i}(t,q^{k})\frac{\partial }{\partial q^{i}}. \end{aligned}$$

(4.21)

This gives

$$\begin{aligned} X_{\mathrm{GR}}=(3c_{1}t+c_{3})\partial _{t}+c_{1}a\partial _{a} \end{aligned}$$

(4.22)

The corresponding generators of a Noether point symmetry are

$$\begin{aligned} X_{1}= & {} \partial _{t},\nonumber \\ X_{2}= & {} 3t\partial _{t}+a\partial _{a} \end{aligned}$$

(4.23)

The simple commutative algebra of the two symmetry generators is

$$\begin{aligned}{}[X_{1},X_{2}]={3X_{1}}. \end{aligned}$$

(4.24)

Since it admits the Noether symmetry $\partial _{t}$ ,the first corresponding conserved quantity ($I_{1}$) related to the energy conservation via the time translation symmetry is

$$\begin{aligned} I_{1,{\mathrm{GR}}}=E_{\mathcal{L}_{\mathrm{GR}}}={\mathcal {H}}_{\mathrm{GR}}=0. \end{aligned}$$

(4.25)

The second Noether integral is

(4.26)

Here we set ${B(t,a,\phi )=B_{(t)}+B_{(a,\phi )}}$ by assuming $B_{(t)}=0$ in the simplest case.

Using Eq. (3.3), i.e. ${\mathcal {H}}_{\mathrm{GR}}=0$, this gives

$$\begin{aligned} I_{2,\mathrm{GR}}= & {} (3c_{1}t+c_{3}){{\mathcal {H}}_{\mathrm{GR}}}-6c_{1}a^{2}{\dot{a}},\nonumber \\= & {} -6c_{1}a^{2}{\dot{a}}. \end{aligned}$$

(4.27)

That leads to

$$\begin{aligned} a(t)= & {} \Big (-\frac{I_{2,\mathrm{GR}}}{2c_{1}}\Big )^{1/3}t^{1/3},\nonumber \\ a(t)= & {} a_{0}t^{1/3}. \end{aligned}$$

(4.28)

Comparing to the scale factor for single-component universe see section 5.3 of Ref. [66],

$$\begin{aligned} a(t)=(\frac{t}{t_{0}})^{\frac{2}{3+3w}}, \end{aligned}$$

(4.29)

this yields the equation of state parameter for dominance role of kinetic part of the scalar field rather than its potential, i.e. ${\dot{\phi }}^{2}\gg V(\phi )$

$$\begin{aligned} w=1, \end{aligned}$$

(4.30)

this indicates the existence a kinetic dominance (KD) or stiff matter phase after the big bang but before the slow roll condition took place. The form of scalar field that corresponds to that scale factor is $\phi \propto t^{1/6}$ [71]

For $\alpha (a)=c_{1}\ln a$ and $\varphi (\phi )=c_{2}\ln \phi$, this gives $\alpha _{\phi }=0$, $\varphi _{a}=\frac{\partial }{\partial a}(c_{2}\ln \phi )=0$ and $c_{2} = \frac{\phi }{a}(\ln a-c_{1})$ as shown as an example in Table 1. The constant of motion and gauge function related to this form of solutions take the form

$$\begin{aligned} \Sigma _{0,GR(2)}= & {} -12c_{1}a{\dot{a}}\ln a+2\epsilon c_{2}a^{3}{\dot{\phi }}\ln {\phi },\nonumber \\ B_{(a,\phi )}= & {} -12c_{1}a{\dot{a}}+2\epsilon c_{2}a^{3}{\dot{\phi }}\ln \phi \,, \end{aligned}$$

(4.31)

respectively. From Eq. (4.20), we find

$$\begin{aligned} \frac{I_{2,GR(2)}}{12c_{1}}t=c_{3}t=\frac{3a^{2}}{4}-\frac{1}{2} a^{2}\ln a \end{aligned}$$

(4.32)

which leads to a scale factor written of the form

$$\begin{aligned} a(t)= \frac{\pm 2\sqrt{|c_{3}|t}}{\sqrt{\text {production} \ln (\frac{4|c_{3}|t}{e^{3}})}}\,. \end{aligned}$$

(4.33)

Here we will not elaborate on this case further.

4.2 NMDC universe

For the NMDC universe, the Lagrangian for NMDC gravity is written as

$$\begin{aligned}&{\mathcal {L}}_{NMDC}\nonumber \\&\quad =\sqrt{-g}\Big [ {R}-(\epsilon g_{\mu \nu }+\kappa G_{\mu \nu })\nabla ^{\mu }\phi \nabla ^{\nu }\phi -2V(\phi )\Big ],\nonumber \\ \end{aligned}$$

(4.34)

in which it can illustrate the point-like Lagrangian of NMDC as

$$\begin{aligned} {\mathcal {L}}_{NMDC}= & {} {-6a{\dot{a}}^2}+\bigg ({\epsilon }{a^3{\dot{\phi }}^2}-3\kappa a{\dot{a}}^2 {\dot{\phi }}^2 \bigg ) \nonumber \\&-2a^3 V(\phi )-2a^3\rho _{m}(a). \end{aligned}$$

(4.35)

The Noether gauge condition in this case yields

$$\begin{aligned} X^{[1]}_{NGS}{\mathcal {L}}_{NMDC}+{\mathcal {L}}_{NMDC} D_{t}\tau =D_{t}B \end{aligned}$$

(4.36)

Using the Noether gauge condition to ${\mathcal {L}}_{NMDC}$, this gives 34 terms which can be expressed as follows:

$$\begin{aligned} D_{t}B= & {} -6a^{2}V(\phi )\alpha -6a^{2}\alpha \rho _{m}(a)-6\alpha {\dot{a}}^{2}-2a^{3}\varphi V'(\phi )\nonumber \\&+3\epsilon a^{2}\alpha {\dot{\phi }}^{2}-3\kappa \alpha {\dot{a}}^{2}{\dot{\phi }}^{2}-2a^{3}\alpha \rho '_{m}\nonumber \\&-12a{\dot{a}}\alpha _{t}-6\kappa a{\dot{a}}{\dot{\phi }}^{2}\alpha _{t}-2a^{3}V(\phi )\tau _{t}-2a^{3}\rho _{m}(a)\tau _{t} \nonumber \\&+6a{\dot{a}}^{2}\tau _{t}-\epsilon a^{3}{\dot{\phi }}^{2}\tau _{t}+9\kappa a {\dot{a}}^{2}{\dot{\phi }}^{2}\tau _{t}\nonumber \\&+2\epsilon a^{3}{\dot{\phi }}\varphi _{t}-6\kappa a{\dot{a}}^{2}{\dot{\phi }}\varphi _{t}-12a{\dot{a}}{\dot{\phi }}\alpha _{\phi }-6\kappa a{\dot{a}}{\dot{\phi }}^{3}\alpha _{\phi }\nonumber \\&-2a^{3}V(\phi ){\dot{\phi }}\tau _{\phi }-2a^{3}\rho _{m}(a){\dot{\phi }}\tau _{\phi }\nonumber \\&+6a{\dot{a}}^{2}{\dot{\phi }}\tau _{\phi }-\epsilon a^{3}{\dot{\phi }}^{3}\tau _{\phi }+9\kappa a{\dot{a}}^{2}{\dot{\phi }}^{3}\tau _{\phi }+2\epsilon a^{3}{\dot{\phi }}^{2}\varphi _{\phi }\nonumber \\&-6\kappa a {\dot{a}}^{2}{\dot{\phi }}^{2}\varphi _{\phi }-12a{\dot{a}}^{2}\alpha _{a}-6\kappa a{\dot{a}}^{2}{\dot{\phi }}^{2}\alpha _{a}\nonumber \\&-2a^{3}V(\phi ){\dot{a}}\tau _{a}-2a^{3}\rho _{m}(a){\dot{a}}\tau _{a}+ 6a{\dot{a}}^{3}\tau _{a}-\epsilon a^{3}{\dot{a}}{\dot{\phi }}^{2}\tau _{a}\nonumber \\&+9\kappa a{\dot{a}}^{3}{\dot{\phi }}^{2}\tau _{a}+2\epsilon a^{3}{\dot{a}}{\dot{\phi }}\varphi _{a}-6\kappa a{\dot{a}}^{3}{\dot{\phi }}\varphi _{a}. \end{aligned}$$

(4.37)

After separation of polynomials and monomials, we can express each term in a more compact form as follows:

$$\begin{aligned} {\dot{a}}^{2}\Big [ -6\alpha +6a\tau _{t}-12a\alpha _{a} \Big ]= & {} 0, \end{aligned}$$

(4.38)

$$\begin{aligned} {\dot{\phi }}^{2}\Big [ 3\epsilon a^{2}\alpha -\epsilon a^{3}\tau _{t}+2\epsilon a^{3}\varphi _{\phi }\Big ]= & {} 0,\end{aligned}$$

(4.39)

$$\begin{aligned} {\dot{a}}^{2}{\dot{\phi }}^{2}\Big [ -3\kappa \alpha +9\kappa a\tau _{t}-6\kappa a\varphi _{\phi }-6\kappa a\alpha _{a} \Big ]= & {} 0,\end{aligned}$$

(4.40)

$$\begin{aligned} {\dot{a}}{\dot{\phi }}^{2}\Big [ -6\kappa a\alpha _{t}-\epsilon a^{3}\tau _{a} \Big ]= & {} 0,\end{aligned}$$

(4.41)

$$\begin{aligned} {\dot{a}}^{2}{\dot{\phi }}\Big [ -6\kappa a \varphi _{t}+6a{\tau _{\phi }} \Big ]= & {} 0,\end{aligned}$$

(4.42)

$$\begin{aligned} {\dot{a}}{\dot{\phi }}\Big [ -12a\alpha _{\phi }+2\epsilon a^{3}\varphi _{a}\Big ]= & {} 0,\end{aligned}$$

(4.43)

$$\begin{aligned} {\dot{a}}{\dot{\phi }}^{3}\Big [ -6\kappa a \alpha _{\phi } \Big ]= & {} 0, \end{aligned}$$

(4.44)

$$\begin{aligned} {\dot{\phi }}^{3}\Big [ -\epsilon a^{3}\tau _{\phi } \Big ]= & {} 0, \end{aligned}$$

(4.45)

$$\begin{aligned} {\dot{a}}^{2}{\dot{\phi }}^{3}\Big [ 9\kappa a \tau _{\phi } \Big ]= & {} 0,\end{aligned}$$

(4.46)

$$\begin{aligned} {\dot{a}}^{3}\Big [ 6a\tau _{a} \Big ]= & {} 0, \end{aligned}$$

(4.47)

$$\begin{aligned} {\dot{a}}^{3}{\dot{\phi }}^{2} \Big [ 9\kappa a \tau _{a} \Big ]= & {} 0,\end{aligned}$$

(4.48)

$$\begin{aligned} {\dot{a}}^{3}{\dot{\phi }}\Big [ -6\kappa a \varphi _{a} \Big ]= & {} 0, \end{aligned}$$

(4.49)

$$\begin{aligned} {\dot{a}}B_{a}= & {} {\dot{a}}\Big [ -12a\alpha _{t}-2a^{3}V(\phi )\tau _{a}-2a^{3}\rho _{m}(a)\tau _{a} \Big ],\end{aligned}$$

(4.50)

$$\begin{aligned} {\dot{\phi }}B_{\phi }= & {} {\dot{\phi }}\Big [2\epsilon a^{3}\varphi _{t}-2a^{3}V(\phi )\tau _{\phi }-2a^{3}\rho _{m}(a)\tau _{\phi } \Big ], \end{aligned}$$

(4.51)

$$\begin{aligned} B_{t}= & {} -\alpha \Big [ 6a^{2}V(\phi )+6a^{2}\rho _{m}(a)+2a^{3}\rho '_{m}(a)\big ]{-2\varphi } a^{3}V'(\phi )\nonumber \\&{-\tau _{t}}\big [ 2a^{3}V(\phi )+2a^{3}\rho '_{m}(a) \Big ]. \end{aligned}$$

(4.52)

The constraint equations and the PDEs can be given as follows:

$$\begin{aligned} \varphi _a=\varphi _t=\alpha _\phi =\alpha _t=\tau _a=\tau _{\phi }= & {} 0. \end{aligned}$$

(4.53)

$$\begin{aligned} \alpha +2a\alpha _a-a\tau _t= & {} 0 \end{aligned}$$

(4.54)

$$\begin{aligned} 3\alpha +2a\varphi _\phi -a\tau _t= & {} 0 \end{aligned}$$

(4.55)

$$\begin{aligned} \alpha +2a\varphi _\phi +2a\alpha _a-3a\tau _{t}= & {} 0 \end{aligned}$$

(4.56)

$$\begin{aligned} B_{a}=B_{\phi }= & {} 0 \end{aligned}$$

(4.57)

$$\begin{aligned} B_t= & {} -\alpha \big [ 6a^2V(\phi )+6a^2\rho _m(a) +2a^3\rho '_m(a) \Big ] \nonumber \\&-\varphi \Big [(2a^3 V'(\phi )\Big ]\nonumber \\&-\tau _t \Big [ 2a^3V(\phi )+2a^3\rho _m(a) \Big ] . \end{aligned}$$

(4.58)

What is different from GR is the non-vanishing of two terms contributing to NMDC, i.e. $B_{a}=B_{\phi }=0.$ Offsetting this, Eq. (4.56) is modified due to the existence of the NMDC term, i.e.,

$$\begin{aligned} \kappa {\dot{a}}^{2}{\dot{\phi }}^{2}\Big ( 3\alpha +6a\varphi _{\phi }+6a\alpha _{a}-9a\tau _{t}\Big )=0. \end{aligned}$$

(4.59)

Substituting Eq. (4.54) into Eq. (4.56), this yields the relation $\varphi _{\phi }=\tau _{t}.$ From Eqs. (4.54) and (4.55), this gives

$$\begin{aligned} {-3\alpha =a(t)\varphi _{\phi }={3a(t)(2\alpha _{a}-\tau _{t})}.} \end{aligned}$$

(4.60)

Hence we have one more relation given by

$$\begin{aligned} \varphi _{\phi }=3(2\alpha _{a}-\tau _{t}). \end{aligned}$$

(4.61)

Using the fact that $\varphi _{\phi }=\tau _{t}$, this leads to the relation of $\tau _{t}$ , $\alpha _{a}$ and $\varphi _{\phi }$ as follows:

$$\begin{aligned} \alpha _{a}= & {} \frac{2}{3}\tau _{t}=\frac{2}{3}\varphi _{\phi }. \end{aligned}$$

(4.62)

Substituting Eq. (4.62) into Eq. (4.54), Eq. (4.55), and Eq. (4.56), interestingly this gives the same form of the compulsory condition expressed below:

$$\begin{aligned} \alpha (a)+\frac{a}{3}\tau _{t}=0. \end{aligned}$$

(4.63)

Given $\alpha (a)= c_{1}a$, this gives $\tau (t)=-3c_{1}t+c_{2}$ and $\varphi (\phi )=-3c_{1}\phi + c_{3}$. It is straightforward to show that for the NMDC case we have

$$\begin{aligned} X_{\mathrm{NMDC}}=(-3c_{1}t+c_{2})\partial _{t}+c_{1}a\partial _{a}+(-3c_{1}\phi +c_{3})\partial _{\phi }.\nonumber \\ \end{aligned}$$

(4.64)

Notice that it is a coincidence that the Noether vector as shown in Ref. [74] is the same form as that given in Eq. (4.64). However, the constant factors used between those two works are slightly different. With the condition that shows in Eq. (4.62) causing us to set further that $c_{1}=0$. Therefore, we can define the forms of $\alpha , \varphi (\phi )$ and $\tau (t)$ in the simplest form as follow:

$$\begin{aligned} \alpha= & {} 0, \end{aligned}$$

(4.65)

$$\begin{aligned} \varphi= & {} c_{3},\end{aligned}$$

(4.66)

$$\begin{aligned} \tau= & {} c_{2}. \end{aligned}$$

(4.67)

The constant of motion for NMDC case reads

(4.68)

(4.69)

A first integral of the system or a conserved quantity associated with $X_{NGS}$ can be derived from Eq. (4.19) where in the NMDC case we have used $\xi =\tau = c_{2}$ and $\eta ^{i}=\{\alpha =0,\varphi =c_{3}\}.$ The Lagrangian-related Noether vector X for NMDC is

$$\begin{aligned} X_{\mathrm{NMDC}}= c_{2}\partial _{t}+c_{3}\partial _{\phi }. \end{aligned}$$

(4.70)

The corresponding Noether symmetries are

$$\begin{aligned} X_{1}= & {} c_{2}\partial _{t},\nonumber \\ X_{2}= & {} c_{3}\partial _{\phi }. \end{aligned}$$

(4.71)

The simple commutative algebra of the two symmetry generators is

(4.72)

(4.73)

where we have assumed that $B(t)_{\mathrm{NMDC}}\equiv B_{1,2}(t)_{\mathrm{NMDC}}= B_{(a,\phi )}+B_{(t)}$. Two ansatz forms of the gauge function are selected in this work, i.e. $B_{1}(t)_{\mathrm{NMDC}}=c_{4}t+c_{5}$ and $B_{2}(t)_{\mathrm{NMDC(2)}}=c_{4}e^{-\lambda t}+c_{5}$ where $\lambda$ is defined in Eq. (4.81). Due to the fact that $B_{a}=B_{\phi }=0$ as shown in Eq. (4.57), it is naturally to set $B_{(a,\phi )}=c_{5}$. Eq. (4.72) may shed some light on the interplay between $\phi (t)$ and a(t). From the fact that

$$\begin{aligned} \frac{d}{dt}\Sigma _{0,\mathrm{NMDC}}=\frac{d}{dt}( I_{2, \mathrm{NMDC}}+ B_{\mathrm{NMDC}})=0, \end{aligned}$$

(4.74)

and the exponential cosmological solution expressed in term of

$$\begin{aligned} a(t)= & {} e^{\frac{1}{2}H_{0}t}, \end{aligned}$$

(4.75)

$$\begin{aligned} {\dot{a}}(t)= & {} \frac{1}{2}H_{0}e^{\frac{1}{2}H_{0}t}, \end{aligned}$$

(4.76)

where $H_{0}$ is a constant [72, 73]. This leads to an interesting arrangement shown below. For $B_{1}(t)_{\mathrm{NMDC}}=c_{4}t+c_{5}$, this gives

$$\begin{aligned} \Big [\frac{c_{4}t+c_{5}+I_{2,\mathrm{NMDC}}}{2\epsilon c_{3}}\Big ]e^{-\frac{3}{2}H_{0}t}-\frac{d\phi }{dt}\Big (1-\frac{3\kappa H_{0}^{2}}{4\epsilon }\Big )=0.\nonumber \\ \end{aligned}$$

(4.77)

Instead of using the relations $\frac{dt}{\tau }=\frac{da}{\eta _{a}(a)}=\frac{d\phi }{\varphi (\phi )}$ as shown in Eq. (35) of Ref. [74] to get a power-law form of a scale factor and the linear forms of scalar field, see p.12 of Ref. [74], we have assumed an exponential form of the scale factor, $a(t)=e^{\frac{1}{2}H_{0}t}$, to quantify the accelerating expansion of the universe and in order to examine the possible forms of the scalar field under the linear form of gauge function, i.e., $B_{1}(t)_{\mathrm{NMDC}}=c_{4}t+c_{5}$, via the Rund–Trautman identity that plays the central role in the Noether’s theorem. The contribution of $\kappa$ leads to the linear part of the solution of Eq. (4.77) as shown below

$$\begin{aligned} \phi (t)=(c_{6}+c_{7}t)e^{-\lambda t}+C, \end{aligned}$$

(4.78)

where

$$\begin{aligned} c_{6}\equiv & {} -\frac{\Big (2c_{4}+3H_{0}(c_{5}+I_{2,{\mathrm{NMDC}}})\Big )}{9\epsilon c_{3}H_{0}^{2}(1-\frac{3\kappa H_{0}^{2}}{4\epsilon })}, \end{aligned}$$

(4.79)

$$\begin{aligned} c_{7}\equiv & {} -\frac{c_{4}}{3\epsilon c_{3}H_{0}(1-\frac{3\kappa H^{2}_{0}}{4\epsilon })}, \end{aligned}$$

(4.80)

$$\begin{aligned} \lambda\equiv & {} \frac{3H_{0}}{2}, \end{aligned}$$

(4.81)

$$\begin{aligned} C= & {} \text { an arbitrary constant.} \end{aligned}$$

(4.82)

For $B_{2}(t)_{\mathrm{NMDC}}=c_{4}e^{-\lambda t}+c_{5}$ , this gives

$$\begin{aligned} \Big [\frac{c_{4}e^{-\lambda t}+c_{5}+I_{2,{\mathrm{NMDC}}}}{2\epsilon c_{3}}\Big ]e^{-\lambda t}-\frac{d\phi }{dt}\Big (1-\frac{3\kappa H_{0}^{2}}{4\epsilon }\Big )=0.\nonumber \\ \end{aligned}$$

(4.83)

The solution of Eq. (4.83) is

$$\begin{aligned} \phi (t)=({\tilde{c}}_{6}+{\tilde{c}}_{8}t)e^{-2\lambda t}+{\tilde{c}}_{7}e^{-\lambda t}+C[1], \end{aligned}$$

(4.84)

where

$$\begin{aligned} {\tilde{c}}_{6}= & {} -c_{4}/18\epsilon c_{3}H_{0}^{2}(1-\frac{3\kappa H_{0}^{2}}{4\epsilon }), \end{aligned}$$

(4.85)

$$\begin{aligned} {\tilde{c}}_{7}= & {} -\frac{(c_{5}+I_{2,NMDC})}{3\epsilon c_{3}(1-\frac{3\kappa H_{0}^{2}}{4\epsilon })}, \end{aligned}$$

(4.86)

$$\begin{aligned} {\tilde{c}}_{8}= & {} -c_{4}/6\epsilon c_{3}H_{0}^{2}(1-\frac{3\kappa H_{0}^{2}}{4\epsilon }). \end{aligned}$$

(4.87)

5 Conclusion

In the present work, we have considered a formal framework of NMDC gravity and derived the point-like Lagrangian for underlying theory. We have studied a Hessian matrix and quantified the Euler–Lagrange equations and Hamiltonian equations of the Einstein (GR) and the NMDC universes. We further discussed the NGS approach for the GR and NMDC universe. We have studied exact cosmological solutions of both theories with the help of the Noether symmetries of point-like Lagrangian.

We have assumed the linear forms of $\alpha (a),\varphi (\phi )$ and $\tau (t)$ that are compatible with the structure of the standard GR and explained the possible emergence of the kinetic field dominated phase which might take place before the inflationary stage. We discovered that the NGS condition under the structure of NMDC gravity can eliminate the dependence of the variables a(t) and $\phi (t)$ on a gauge function. Consequently, a gauge function is at most dependent on the time variable only. Two ansatz forms of B(t) are chosen as $B_{1}(t)_{\mathrm{NMDC}}=c_{4}t+c_{5}$ and $B_{2}(t)_{\mathrm{NMDC}}=c_{4}e^{-\lambda t}+c_{5}$. Interestingly, we have found the new form of the solutions of $\phi (t)=(c_{6}+c_{7}t)e^{-\lambda t}+C$ for the first case and $\phi (t)=({\tilde{c}}_{6}+{\tilde{c}}_{8}t)e^{-2\lambda t}+{\tilde{c}}_{7}e^{-\lambda t}+C$ for the second case. However, it is very interesting to figure out the physical consequences of the found solutions as shown in Eqs. (4.78) and (4.84) that we leave this for further investigation. Finally, it is hoped that this work will stimulate further research on seeking other possible solutions of the system equations based on Noether gauge symmetry as shown in Eqs. (4.10–4.16) for GR and Eqs. (4.53–4.58) for NMDC cosmology.