# Help in solving Quintessential scalar field using Steep Potential in cosmology

I am attempting to solve the differential equation $$\ddot\phi + 3H\dot\phi + \dfrac{dV}{d\phi} = 0.$$

For $$V(\phi) = V_{0}e^{-\lambda\phi}$$, where $$V_{0} = 0.7$$, $$\lambda = 0.1$$ and $$V'(\phi) = \dfrac{dV}{d\phi}$$

# Attempt to solve second-order ODE.

def DSDX(S,T):
phi,deri_phi = S


# Declaring some important constants required in the program.

omega_r0 = 8.24e-5
omega_m0 = 0.27

lamb = 0.1


# Defining function for $$V(\phi)$$ and $$V'(\phi)$$

def V(x,y):
return 0.7*np.exp(-1*x*y)

def deri_V(x,y):
return -1*x*V(x,y)


Now, my first task is to make the differential equation dimensionless, thus needed variable $$T = \log_{e}(1+z)$$ and some calculation will end up getting equation.

Potential used $$V(\phi) = V_{0}e^{-\lambda\phi}$$ Potential derivative $$V'(\phi) = -\lambda V(\phi)$$ Normalised Hubble Parameter defined $$E^2 = \dfrac{\Omega_{m0}.e^{3T} + \Omega_{r0}.e^{-4T} + V(\phi)}{1 - \dfrac{1}{6}\left(\dfrac{d\phi}{dT}\right)^2}$$ Derivative of Normalised Hubble Parameter $$E' = -\dfrac{\Omega_{r}}{2E} - 1.5E\left( 1 + \dfrac{1}{6}\left(\dfrac{d\phi}{dT}\right)^2\right) + 1.5\dfrac{V(\phi)}{E}$$

# Factors Required.

E_squared = ((omega_m0)*np.exp(3*T) + (omega_r0)*np.exp(4*T) + V(lamb,phi))/(1-(deri_phi**2)/6)
E = np.sqrt(np.abs(E_squared))
deri_E = (-0.5/E)*omega_r0*np.exp(4*T) - (1.5)*E*(1+((deri_phi**2)/6)) + 1.5*V(lamb,phi)/E


# Function Return.

return [deri_phi, 9*deri_phi - 3*deri_E*deri_phi/E - 3*deri_V(lamb,phi)/E_squared]


# Initial conditions

phi0 = 0.1
deri_phi0 = 1e-5
s0 = [phi0,deri_phi0]

Time = np.log(1+np.linspace(1e8,0,10000))
sol = odeint(DSDX,s0,Time)


# Plotting graph

phi , deri_phi = sol.T
plt.title('Evolution of Scalar Field $$\phi$$ for $$\lambda = 0.1$$')
plt.plot(Time,phi,color = 'red')
plt.xlabel('$$T$$')
plt.ylabel('$$\phi(T)$$')

plt.grid()