Tutorial: FLRW Cosmology

This tutorial demonstrates a particular example of the symbolic tools in pystein. For other tutorials see all tutorials.

The pystein.metric module has several well-known metrics predefined. We will use the Friedman-Lemaitre-Roberston-Walker (FLRW) metric for this tutorial:

from pystein import metric, gravity
from pystein import utilities

flrw = metric.flrw().subs({'c': 1})
flrw

$\large \displaystyle - c^{2} \operatorname{d}t \otimes \operatorname{d}t + a^{2}{\left(t \right)} \left(\operatorname{d}x \otimes \operatorname{d}x + \operatorname{d}y \otimes \operatorname{d}y + \operatorname{d}z \otimes \operatorname{d}z\right)$

We can use the gravity module to compute the einstein equation components:

efe_00 = utilities.full_simplify(gravity.einstein_equation(0, 0, flrw))
efe_00

$\large \displaystyle \frac{3 \left(\frac{d}{d t} a{\left(t \right)}\right)^{2}}{a^{2}{\left(t \right)}} = 0$

The metric module contains a utility for simplifying the notation used for derivatives in sympy, which we’ll use to recover the familiar Friedmann equation:

metric.simplify_deriv_notation(efe_00, flrw, use_dots=True)

$\large \displaystyle \frac{3 \dot{a}^{2}{\left(t \right)}}{a^{2}{\left(t \right)}} = 0$