# Which is the total energy density constraint for the gravitational wave background?

Different constraints from different measuring methods have been set on different frequency ranges of the energy density of gravitational waves, as shown in the picture below. What is the total energy density constraint?

Nelson Christensen, "Stochastic Gravitational Wave Backgrounds", [https://arxiv.org/pdf/1811.08797.pdf]

• For some of these observations (for example the "indirect limits", based on BBN and recombination), you get a constraint on the total energy density, and for others (eg LIGO) you get a constraint on the spectral energy density (energy per logarithmic frequency interval). Which observations do you want to know about specifically? Aug 23, 2021 at 15:29
• The constraint on the total energy density. But how can just one observation constraint the total energy density if this is composed of all the frequency range? Aug 23, 2021 at 15:48
• "The constraint on the total energy density" from which observation? Aug 23, 2021 at 15:50
• Cant we have a constraint on total energy density from the combination of all observations? Aug 23, 2021 at 15:54
• OK. So the only total energy density constraints come from the BBN and recombination observations (what's called "indirect limits" on the plot in your question). These limit $\Omega_{\rm GW} < 3.8 \times 10^{-6}$, from Section I.D of arxiv.org/abs/1511.05994. There is probably a more up-to-date number, but it won't have changed by very much. Aug 23, 2021 at 16:04

First, some notation. I will use units where $$c=1$$.

The energy density in gravitational waves is $$$$\rho_{\rm GW} = \frac{1}{32 \pi G_N} \left(\langle \bar{h}_+ ^2 \rangle + \langle \bar{h}_\times ^2 \rangle\right)$$$$ where $$\langle h_+^2\rangle$$ is the average over several wavelengths of the squared amplitude of the "plus" polarization of the gravitational wave, and $$\langle h_\times^2\rangle$$ is the same for the "cross polarization".

We frequently define $$\Omega_{\rm GW}$$ as the ratio of the energy density to the energy density $$\rho_{\rm GW}=3 H_0^2/(8\pi G_N)$$ (where $$H_0$$ is the Hubble constant) needed to have a spatially flat Universe $$$$\Omega_{\rm GW} = \frac{\rho_{\rm GW}}{\rho_c} = \frac{8 \pi G_N \rho_{\rm GW}}{3 H_0^2}$$$$ The spectral energy density $$\Omega_{\rm GW}(f)$$ is the energy density per logarithmic frequency interval $$$$\Omega_{\rm GW}(f) = f \frac{{\rm d} \Omega_{\rm GW}}{{\rm d} f}$$$$ where $${\rm d} \Omega_{\rm GW}$$ is the amount of energy density (normalized by $$\rho_c$$) in the frequency interval $$f$$ to $$f+{\rm d} f$$.

The are two kinds of constraints on the gravitational-wave background.

• Frequency-dependent constraints, for example the constraints placed by LIGO, LISA, and the CMB polarization. These do not measure the total energy density in gravitational waves $$\Omega_{\rm GW}$$, but rather the spectral energy density $$\Omega_{\rm GW}(f)$$. These constraints tend to be "direct" measurements of gravitational waves; a given detector is only sensitive to a range of frequencies of gravitational waves. You can convert this into a constraint on $$\Omega_{\rm GW}$$ in a given frequency band, if you are willing to assume a model for $$\Omega_{\rm GW}(f)$$. The curves you have plotted are power-law integrated curves (PI curve) [1] -- any power-law background which is tangent to a PI curve is ruled out at the 2-$$\sigma$$ level.

• Frequency-integrated constraints, such as the ones from BBN and CMB recombination. These constraints tend to be "indirect" -- both of these constraints come from constraints on the effective number of relativistic degrees of freedom during BBN and CMB recombination. If there was a significant energy density present in gravitational waves during these periods, it would change the expansion history (the scale factor $$a(t)$$), and would run afoul of observations of these epochs. Therefore, these constraints are constraints on the total, integrated $$\Omega_{\rm GW}$$, not $$\Omega_{\rm GW}(f)$$. Ref [2] gives a constraint $$\Omega_{\rm GW}<3.8 \times 10^{-6}$$ from these types of measurements; since this reference is from 2015, there is probably an updated number, but I suspect it is not much stronger. As you can see from the plot, the indirect constraints are not necessarily the strongest constraints in any given frequency interval.

Finally, two updates to the plot you have shown:

• NanoGRAV may have detected the gravitational-wave background in the nHz region using pulsar timing arrays [3]. It's too early to say so far, but their data are consistent with a common red process. The smoking gun signature would be an observation of the Hellings and Downs curve, which come from correlations between pulsars; this has not yet been observed.

• The most recent LIGO-Virgo constraints come from earlier this year [4], and are a factor of a few better than the ones on your plot.

References

• One last question. That indirect contraint from [2] is a density constraint on the actual density or on the density at the time of recombination? Aug 25, 2021 at 21:58
• @Manuel It's a combination of two constraints. One constrains the expansion rate (and hence the density, or more technically "the effective number of relativistic degrees of freedom") at big bang nucleosynthesis, and the other constrains the expansion rate, or equivalently density, at CMB recombination. Aug 26, 2021 at 0:33

#Andrew Nice Stuff! I'm just getting started with GR and I'm especially interested in the grav wave background radiation from the development of the universe. At the lowest freqs I see an almost thermal spectrum much like the spectrum of the CMB. This is the part I'd like to focus.my studies on first. But I don't understand the units on the vertical axis of the stochastic background curves in the graph. Often, the CMB is quoted as a (quasi?) Thermal bath at 3 K. Freq spectrum About 10-500 GHz [edited since posting]