Relating gravitational wave constraints from primordial nucleosynthesis, pulsar timing, laser interferometers, and the CMB: implications for the early universe

TL;DR

The paper derives a general master equation linking the CMB-observable tensor-to-scalar ratio $r$ to the present-day gravitational-wave spectrum $\Omega_{0}^{\rm gw}(f)$, encapsulating early-universe physics in two model-independent averages, $\hat{w}(f)$ and $\hat{n}_{t}(f)$. These quantities summarize the effective equation of state and tensor tilt across the primordial dark age, allowing robust, model-insensitive constraints on stiff-energy components prior to BBN. By examining how $r$ and $\Omega_{0}^{\rm gw}(f)$ constrain $\hat{w}(f)$ and $\hat{n}_{t}(f)$, the work shows that a future detection of $r$ would yield a remarkably tight bound on the early-universe equation of state, $\hat{w}_{\rm end}$, independent of microscopic inflationary details. The framework unifies constraints across CMB, pulsar timing, and laser-interferometer scales and highlights powerful consistency checks and sBBN-imposed limits, with significant implications for probing the primordial dark age and any stiff-energy components.

Abstract

We derive a general master equation relating the gravitational-wave observables r and Omega_gw(f). Here r is the tensor-to-scalar ratio, constrained by cosmic-microwave-background (CMB) experiments; and Omega_gw(f) is the energy spectrum of primordial gravitational-waves, constrained e.g. by pulsar-timing measurements, laser-interferometer experiments, and Big Bang Nucleosynthesis (BBN). Differentiating the master equation yields a new expression for the tilt d(ln Omega_gw(f))/d(ln f). The relationship between r and Omega_gw(f) depends sensitively on the uncertain physics of the early universe, and we show that this uncertainty may be encapsulated (in a model-independent way) by two quantities: w_hat(f) and nt_hat(f), where nt_hat(f) is a certain logarithmic average over nt(k) (the primordial tensor spectral index); and w_hat(f) is a certain logarithmic average over w_tilde(a) (the effective equation-of-state in the early universe, after horizon re-entry). Here the effective equation-of-state parameter w_tilde(a) is a combination of the ordinary equation-of-state parameter w(a) and the bulk viscosity zeta(a). Thus, by comparing constraints on r and Omega_gw(f), one can obtain (remarkably tight) constraints in the [w_hat(f), nt_hat(f)] plane. In particular, this is the best way to constrain (or detect) the presence of a ``stiff'' energy component (with w > 1/3) in the early universe, prior to BBN. Finally, although most of our analysis does not assume inflation, we point out that if CMB experiments detect a non-zero value for r, then we will immediately obtain (as a free by-product) a new upper bound w_hat < 0.55 on the logarithmically averaged effective equation-of-state parameter during the ``primordial dark age'' between the end of inflation and the start of BBN.

Figures (7)

• Figure 1: How the components of the cosmological energy budget scale with cosmological expansion: "stiff energy" (solid purple line), radiation (long-dashed red line), matter (dotted blue line), and dark energy (dot-dashed green line). Components with higher $w$ tend to dominate at earlier times. Our universe may be dominated by a "stiff energy" component (with $w>1/3$) prior to Big Bang Nucleosynthesis (but after inflation).
• Figure 2: This figure relates to "Case 1," discussed in Sec. \ref{['Case_1']}. The curves show the upper bound on $\Omega_{0}^{{\rm gw}}(f)$, over the range $f_{{\rm cmb}}^{}<f<f_{{\rm end}}^{}$, for various assumed values of $\hat{w}_{{\rm max}}^{}$ and $\hat{n}_{t,{\rm max}}^{}$. The 4 solid black curves correspond (from bottom to top) to $\hat{w}_{{\rm max}}^{}=\{1/3, 0.4, 0.5, 0.6\}$ and $\hat{n}_{t,{\rm max}}^{}=0$. The 4 dotted red curves show the same thing, but now with $n_{t,{\rm max}}^{}=-r_{{\rm max}}^{}/8$. The 4 dashed blue curves correspond (from bottom to top) to $\hat{n}_{t,{\rm max}}^{}=\{0.1,0.2,0.3,0.4\}$ and $\hat{w}_{{\rm max}}^{}=1/3$. Note that frequencies below $f_{c}^{} =f_{{\rm bbn}}^{}\approx 10^{-11}~{\rm Hz}$ re-entered the Hubble horizon after BBN, and hence are unaffected by assumptions about $\hat{w}$ during the primordial dark age. The current and future experimental constraints shown in the figure are discussed in the text, at the end of Sec. \ref{['Case_1']}.
• Figure 3: Bounds from combining CMB and LI/PT experiments. (This figure relates to Cases 2, 3, and 4, discussed in Secs. \ref{['Case_2']}, \ref{['Case_3']}, and \ref{['Case_4']}.) We show examples of the constraints in the $\{\hat{w}(f), \hat{n}_{t}^{}(f)\}$ plane that follow from CMB constraints on $r$ and LI/PT constraints on $\Omega_{0}^{{\rm gw}}(f)$. In both the top and bottom panels, the 4 curves correspond to: $f_{{\rm LIGO}}^{}=100~{\rm Hz}$ (red dotted); $f_{{\rm BBO}}^{}=0.3~{\rm Hz}$ (green dashed); $f_{{\rm LISA}}^{}=10^{-3}~{\rm Hz}$ (grey solid); and $f_{{\rm pulsar}}^{}=10^{-9}~{\rm Hz}$ (black dot-dashed). In the top panel, all 4 curves are plotted assuming $\Omega_{0}^{{\rm gw}}(f)/r=10^{-7}$. So, for example, suppose CMB and LI/PT experiments find: (i) $\Omega_{0}^{{\rm gw}}(f)=10^{-8}$ and $r=0.1$ (Case 2); or (ii) $\Omega_{0}^{{\rm gw}}(f)=10^{-8}$ and $r<0.1$ (Case 3); or (iii) $\Omega_{0}^{{\rm gw}}(f)<10^{-8}$ and $r=0.1$ (Case 4). Then $\{\hat{w}(f),\hat{n}_{t}^{}(f)\}$ must lie: (i) on, (ii) above, or (iii) below the respective curve. The bottom panel is similar, but instead of all curves corresponding to $\Omega_{0}^{{\rm gw}}(f)/r=10^{-7}$, we take $\Omega_{0}^{{\rm gw}}(f)/r$ to be closer to the minimum possible value for each experiment: $10^{-8}$ (at $f_{{\rm LIGO}}^{}$); $10^{-10}$ (at $f_{{\rm LISA}}^{}$); $10^{-16}$ (at $f_{{\rm BBO}}^{}$), and $10^{-10}$ (at $f_{{\rm pulsar}}^{}$).
• Figure 4: Bounds from combining CMB and LI/PT experiments. (This figure relates to Cases 3 and 4, discussed in Secs. \ref{['Case_3']} and \ref{['Case_4']}.) In both the top and bottom panels, the 4 curves correspond to the 4 frequencies: $f_{{\rm LIGO}}^{}=100~{\rm Hz}$ (red dotted); $f_{{\rm BBO}}^{}=0.3~{\rm Hz}$ (green dashed); $f_{{\rm LISA}}^{}=10^{-3}~{\rm Hz}$ (grey solid); and $f_{{\rm pulsar}}^{}=10^{-9}~{\rm Hz}$ (black dot-dashed). This figure has 2 interpretations. In Case 3, where LI (or PT) experiments detect $\Omega_{0}^{{\rm gw}}(f)$ and CMB experiments obtain an upper bound $r_{{\rm max}}^{}$, the "bottom" and "left" axis labels apply, and the curves represent $\hat{w}_{{\rm min}}^{}(f)$ (top panel, with the standard inflationary assumption $\hat{n}_{t,{\rm max}}^{}=0$) and $\hat{n}_{t,{\rm min}}^{}(f)$ (bottom panel, with the standard primordial-dark-age assumption $\hat{w}_{{\rm max}}^{}=1/3$), so the actual values of $\hat{w}(f)$ and $\hat{n}_{t}^{}(f)$ lie above the curves. In Case 4, where CMB experiments detect $r$ and LI (or PT) experiments obtain an upper bound $\Omega_{0,{\rm max}}^{{\rm gw}}(f)$, the "top" and "right" axis labels apply, and the curves represent $\hat{w}_{{\rm max}}^{}(f)$ (top panel, with the standard inflationary assumption $\hat{n}_{t}^{}\approx0$) and $\hat{n}_{t,{\rm max}}^{}(f)$ (bottom panel, with the standard primordial-dark-age assumption $\hat{w}(f)\approx1/3$), so the actual values of $\hat{w}(f)$ and $\hat{n}_{t}^{}(f)$ lie below the curves.
• Figure 5: Bound from combining sBBN and CMB constraints. If CMB experiments detect $r$, then the sBBN gravitational-wave constraint immediately requires $\{\hat{w}(f_{{\rm end}}^{}), \hat{n}_{t}^{}(f_{{\rm end}}^{})\}$ to lie below the curves shown in the figure. From top to bottom, the curves correspond to: $r=10^{-3}$ (black dotted curve), $r=10^{-2}$ (purple dot-dashed curve), and $r=10^{-1}$ (green solid curve). Note that the curves are very insensitive to $r$: they hardly move as $r$ varies over the range in which it can be realistically detected by CMB polarization experiments ($10^{-3}<r<10^{-1}$). The horizontal and vertical dashed lines point out that, for $r=\{10^{-1},10^{-2},10^{-3}\}$, respectively: (a) if $\hat{n}_{t}^{}(f)$ is assumed to take its "standard"' value ($\approx 0$), then $\hat{w}(f_{{\rm end}}^{})\lesssim \{0.54,0.57,0.60\}$; and (b) if $\hat{w}(f)$ is assumed to take its "standard" value ($\approx 1/3$), then $\hat{n}_{t}^{} (f_{{\rm end}})\lesssim\{0.36,0.40,0.43\}$.