Cosmology - Fluid Equation

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In this chapter, we will discuss Fluid Equation and how it tells us regarding the density of universe that changes with time.

Estimating ρ_c and ρ in the Present Universe

For the present universe −

$$\rho_c \simeq 10^{11}M_\odot M_{pc}^{-3} \simeq 10\: hydrogen \: atoms \: m^{-3}$$

There is a whole range of critical density in our outer space. Like, for intergalactic medium $\rho_c$ is 1 hydrogen atom $m^{-3}$, whereas for molecular clouds it is $10^6$ hydrogen atoms $m^{−3}$.

We must measure $\rho_c$ considering proper samples of space. Within our galaxy, the value of $\rho_c$ is very high, but our galaxy is not a representative of the whole universe. So, we should go out to space where cosmological principle holds, i.e., distances ≈ 300 Mpc. Looking at 300 Mpc means looking 1 billion years back, but it is still the present universe.

Surveys like SDSS are conducted to determine the actual matter density. They take a 5×500×5 Mpc³ volume, count the number of galaxies and add all the light coming from these galaxies. Under an assumption that 1 L ≡ 1 M, i.e. 1 solar Luminosity ≡ 1 solar Mass.

We do a light to mass conversion and then we try to estimate the number of baryons based on the visible matter particles present in that volume.

For example,

$$1000L_\odot ≡ 1000M_\odot / m_p$$

Where, m_p= mass of proton.

Then we get roughly the baryon number density $\Omega b ∼= 0.025$. This implies $\rho b = 0.25%$ of $\rho_c$. Different surveys have yielded a slightly different value. So, in the local universe, number density of visible matter is much less than critical density, meaning we are living in an open universe.

Mass with a factor of 10 is not included in these surveys because these surveys account for electromagnetic radiation but not dark matter. Giving, $\Omega_m = 0.3 − 0.4$. Still concludes that we are living in an open universe.

Dark matter interacts with gravity. A lot of dark matter can halt the expansion. We haven’t yet formalized how $\rho$ changes with time, for which we need another set of equations.

Thermodynamics states that −

$$dQ = dU + dW$$

For a system growing in terms of size, $dW = P dV$. Expansion of universe is modelled as adiabatic i.e. $dQ = 0$. So, volume change should happen from change in internal energy dU.

Let us take a certain volume of universe of unit comoving radius i.e. $r_c = 1$. If $\rho$ is the density of material within this volume of space, then,

$$M = \frac{4}{3} \pi a^3r_c^3 \rho$$

$$U = \frac{4}{3}\pi a^3\rho c^2$$

Where, U is the Energy density. Let us find out the change in internal energy with time as the universe is expanding.

$$\frac{\mathrm{d} U}{\mathrm{d} t} = 4 \pi a^2 \rho c^2 \frac{\mathrm{d} a}{\mathrm{d} t} + \frac{4}{3}\pi a^3 c^2\frac{\mathrm{d} \rho}{\mathrm{d} t}$$

Similarly, change in volume with time is given by,

$$\frac{\mathrm{d} V}{\mathrm{d} t} = 4\pi a^2 \frac{\mathrm{d} a}{\mathrm{d} t}$$

Substituting $dU = −P dV$. We get,

$$4\pi a^2(c^2 \rho +P)\dot{a}+\frac{4}{3}\pi a^3c^2\dot{\rho} = 0$$

$$\dot{\rho}+3\frac{\dot{a}}{a}\left ( \rho + \frac{P}{c^2} \right ) = 0$$

This is called the Fluid Equation. It tells us how the density of universe changes with time.

Pressure drops as the universe expands. At every instant pressure is changing, but there is no pressure difference between two points in the volume considered, so, the pressure gradient is zero. Only relativistic materials impart pressure, matter is pressure-less.

Friedmann Equation along with Fluid Equation models the universe.

Points to Remember

• Dark matter interacts with gravity. A lot of dark matter can halt the expansion.

• Fluid Equation tells us how the density of universe changes with time.

• Friedmann Equation along with Fluid Equation models the universe.

• Only relativistic materials impart pressure, matter is pressure-less.