Electronic address<EMAIL_ADDRESS>
# Gravitational Wave Direct Detection Does Not Constrain the Tensor Spectral
Index at CMB Scales

William H. Kinney  [ Department of Physics, University at Buffalo, Buffalo, NY
14260, USA

###### Abstract

I discuss constraints on the power spectrum of primordial tensor perturbations
from a combination of Cosmic Microwave Background (CMB) measurements and the
gravitational wave direct detection experiments LIGO/Virgo and DECIGO. There
are two main points: (1) Inflation predicts an approximately power-law form
for the primordial tensor spectrum, but makes no prediction for its amplitude.
Given that neither Planck nor LIGO/Virgo has actually detected primordial
tensor modes, it is trivially true that no model-independent constraint on the
slope of the tensor power spectrum is possible with current data. (2) CMB and
LIGO/Virgo scales differ by more than 19 orders of magnitude, and 16 for
DECIGO. I show that a power-law extrapolation from CMB to direct detection
frequencies overestimates the amplitude of primordial tensor modes by as much
as two orders of magnitude relative to an ensemble of realistic single-field
inflation models. Moreover, the primordial tensor amplitude at direct
detection scales is mostly uncorrelated with the tensor spectral index at CMB
scales, and any constraint is strongly dependent on the specific form of the
inflationary potential.

##  1 Introduction

Cosmological inflation [Starobinsky(1980), Sato(1981a), Sato(1981b),
Kazanas(1980), Guth(1981), Linde(1982), Albrecht and Steinhardt(1982)]
predicts a nearly scale-invariant power-law spectrum of primordial
gravitational waves [Starobinsky(1979), Mukhanov and Chibisov(1981),
Mukhanov(2003), Linde(1983), Hawking(1982), Hawking and Moss(1983),
Starobinsky(1982), Guth and Pi(1982), Bardeen  _et al._(1983)Bardeen,
Steinhardt, and Turner], extending from superhorizon scales to wavelengths
potentially accessible to future direct detection experiments [Turner(1997),
Ungarelli  _et al._(2005)Ungarelli, Corasaniti, Mercer, and Vecchio,
Cooray(2005), Smith  _et al._(2006a)Smith, Kamionkowski, and Cooray, Smith
_et al._(2006b)Smith, Peiris, and Cooray, Chongchitnan and Efstathiou(2006),
Friedman  _et al._(2006)Friedman, Cooray, and Melchiorri, Kawamura  _et
al._(2011)Kawamura _et al._, Kawamura  _et al._(2020)Kawamura _et al._,
Caligiuri  _et al._(2015)Caligiuri, Kosowsky, Kinney, and Seto, Caprini and
Figueroa(2018)].

The Planck measurement of the Cosmic Microwave Background constrains the
amplitude primordial gravitational wave (or tensor) perturbations on scales
comparable to the current Hubble length,

$a_{0}H_{0}=2.248\times 10^{-4}\ \mathrm{Mpc}^{-1},$ (1)

placing an upper bound on the ratio of primordial gravitational wave
perturbations $P_{T}$ to curvature perturbations $P_{\mathcal{R}}$ (the
tensor/scalar ratio), measured at a pivot scale of $k=0.002\
\mathrm{Mpc}^{-1}$ of

$r_{0.002}\equiv\frac{P_{T}}{P_{\mathcal{R}}}\bigg{|}_{k=0.002\
\mathrm{Mpc}^{-1}}<0.06.$ (2)

This bound is from Planck in combination with the BICEP/Keck measurement of
CMB polarization and Baryon Acoustic Oscillation (BAO) measurements [Aghanim
_et al._(2020)Aghanim _et al._]. (I adopt the Planck best-fit value for the
Hubble constant of $H_{0}=67.4\ \mathrm{km/s/Mpc}$ throughout.) Inflation
predicts the production of primordial gravitational waves with amplitude

$r=16\epsilon,$ (3)

where $\epsilon$ is the first slow roll parameter, defined in terms of the
Hubble parameter as

$\epsilon\equiv\frac{1}{H}\frac{dH}{dN}.$ (4)

$dN\equiv-Hdt$ is the number of e-folds of expansion, $a\left(N\right)\propto
e^{-N}$. The spectrum of primordial gravity waves is a power-law,

$P_{T}=\frac{2H^{2}}{\pi^{2}M_{\mathrm{P}}^{2}}\bigg{|}_{k=aH}\propto
k^{n_{T}},$ (5)

where $k=aH$ indicates evaluation when a wavenumber $k$ exits the horizon
during inflation. In the case of single-field inflation, the tensor spectral
spectral index is also determined by $\epsilon$,

$n_{T}=-r/8=-2\epsilon.$ (6)

Because this consistency relation is a prediction of single-field inflation,
constraining the tensor spectral index $n_{T}$ is of great interest. A
constraint on the slope of the tensor spectrum requires measurement of the
tensor amplitude at multiple scales: direct detection experiments such as
LIGO/Virgo are an ideal candidate for such a measurement, and a number of
papers in the recent literature have derived bounds on $n_{T}$ from Planck in
combination with bounds on the contribution of gravity waves to the number of
relativistic degress of freedom $N_{eff}$ and from LIGO/Virgo [Lasky  _et
al._(2016)Lasky _et al._, Cabass  _et al._(2016)Cabass, Pagano, Salvati,
Gerbino, Giusarma, and Melchiorri, Akrami  _et al._(2020)Akrami _et al._,
Giarè  _et al._(2019)Giarè, Di Valentino, and Melchiorri, Tanin and
Tenkanen(2021)]. Bounds from $N_{eff}$ depend on the integrated density in
gravitational waves over all scales, but bounds from LIGO/Virgo depend on
extrapolation to high frequency. In this paper, I focus on the latter case.

A few general points can be made at the outset: First is that neither Planck
nor LIGO/Virgo has actually detected primordial gravitational waves. Both
measurements produce only upper bounds on the primordial tensor amplitude, and
the signal from slow-roll inflation is expected to be many orders of magnitude
below LIGO/Virgo sensitivity [Aasi  _et al._(2014)Aasi _et al._]. It therefore
follows trivially that no model-independent constraint on $n_{T}$ is possible:
if you do not know the value of a function at any point, quoting a constraint
on its slope is obviously nonsensical. A model-dependent constraint can be
derived, e.g. by assuming a value of $r$ at CMB scales, and then placing a
constraint on $n_{T}$ from bounds on $N_{eff}$ or from the non-detection of
primordial tensor modes by LIGO/Virgo. This latter option, however, requires a
substantial extrapolation. LIGO/Virgo is sensitive to gravitational waves
between roughly 10 and 1000 Hz, with maximum sensitivity at a frequency of
about 100 Hz. Translated into wavenumber, this frequency corresponds to
$k=6.47\times 10^{16}\ \mathrm{Mpc}^{-1}$. Comparing the tensor amplitude from
a CMB pivot scale of $k=0.002\ \mathrm{Mpc}^{-1}$ to LIGO/Virgo scales
therefore requires an extrapolation of more than 19 orders of magnitude! It is
not at all obvious that the assumption of a power-law for the tensor spectrum
– which itself presupposes an inflationary origin for the perturbations – is
even approximately valid over such a large range of scales.

Primordial tensor modes are created during inflation by the redshift of
quantum fluctuations to superhorizon scales, with an amplitude which is
determined by the expansion rate at the time a particular wavenumber crosses
the horizon scale (Eq. 5). It is convenient to express the condition for
horizon crossing in terms of the number of e-folds before the end of inflation
at which a mode exits the horizon. I adopt the usual convention for the number
of e-folds $N$, such that $N=0$ denotes the end of inflation and the onset of
reheating, which $N$ increasing going back in time, and earlier into the
inflationary epoch. Long wavelength modes exit the horizon earlier in
inflation, and short wavelength modes exit the horizon later. Because of this,
we expect that wavelengths corresponding to direct detection scales will exit
the horizon near the end of inflation. For a wavenumber $k$, the number of
e-folds at which it crosses outside the horizon during inflation is

$\displaystyle N_{k}=$
$\displaystyle-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\ln\left(\frac{H_{*}}{H_{e}}\right)+\left(\frac{1+3{\bar{w}}}{2}\right)\left|N_{RH}\right|$
(9)
$\displaystyle+\ln\left(\frac{T_{RH}}{T_{eq}}\right)+\frac{1}{3}\ln\left(\frac{g_{*S}(T_{RH})}{g_{*S}(T_{eq})}\right)$
$\displaystyle+\ln\left(\frac{(aH)_{eq}}{(aH)_{0}}\right),$

where $a_{0}H_{0}$ is the inverse of the comoving horizon size in the current
universe, and subscript $eq$ indicates quantities evaluate at matter/radiation
equality. Here $N_{RH}$ is the number of e-folds of expansion during
reheating, and $\bar{w}$ is the average equation of state during the reheating
period, before the universe becomes radiation-dominated. $H_{*}$ is the value
of the Hubble parameter during inflation when the mode exited the horizon, and
$H_{e}$ is the Hubble parameter at the end of inflation. The factor $g_{*S}$
is the entropy in relativistic degrees of freedom. Taking Planck best-fit
values for the matter and Dark Energy density gives

$\ln\left(\frac{(aH)_{eq}}{(aH)_{0}}\right)=3.839.$ (10)

I assume matter-domination during the reheat period, ${\bar{w}}=0$, so that

$\left(\frac{1+3{\bar{w}}}{2}\right)\left|N_{RH}\right|=-\frac{2}{3}\ln\left(\frac{T_{RH}}{\rho_{e}^{1/4}}\right),$
(11)

where $\rho_{e}$ is the energy density at the end of inflation. For the
purposes of the present discussion, it is sufficient to simplify by taking
$H_{*}\simeq H_{e}$, and $g_{*S}\left(T_{RH}\right)\simeq 100$, so that

$\displaystyle N\left(k\right)=$
$\displaystyle-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\frac{1}{3}\ln\left(\frac{T_{RH}}{10^{15}\
\mathrm{GeV}}\right)$ (13)
$\displaystyle+\frac{2}{3}\ln\left(\frac{\Lambda}{10^{15}\
\mathrm{GeV}}\right)+60.4,$

where $\Lambda\simeq\rho_{e}^{1/4}$ is the energy scale of inflation, and

$T_{eq}=3404T_{0}=9295\ \mathrm{K}=8.01\times 10^{-10}\ \mathrm{GeV}.$ (14)

While this is a simplified expression relative to the most general case, it
will be sufficient for our purpose here, since considering more general
reheating models will only increase the associated uncertainty. The reheat
temperature can be at most equal to the energy density, $T_{RH}\leq\Lambda$,
so there is an upper bound on $N\left(k\right)$,

$N\left(k\right)\leq-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\ln\left(\frac{\Lambda}{10^{15}\
\mathrm{GeV}}\right)+60.4.$ (15)

We can similarly place a lower bound on $N\left(k\right)$ by taking the reheat
temperature to be at least the temperature of nucleosynthesis, $T_{RH}\gtrsim
10^{-1}\ \mathrm{GeV}$, so that

$N\left(k\right)\geq-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\frac{2}{3}\ln\left(\frac{\Lambda}{10^{15}\
\mathrm{GeV}}\right)+48.1.$ (16)

Figure 1 shows upper and lower bounds on $N\left(k_{\mathrm{LIGO}}\right)$
versus inflationary energy scale $\Lambda$, and Fig. 2 shows
$N\left(k_{\mathrm{DECIGO}}\right)$. The first thing to note is that for low
scales of the inflationary energy density, $N\left(k\right)$ actually becomes
negative, which means that those modes never leave the horizon during
inflation, and there is no production of gravitational waves at all! However,
the energy scale of inflation and the tensor/scalar ratio at CMB scales are
directly related,

$\Lambda\simeq r^{1/4}\times\left(3.3\times 10^{16}\ \mathrm{Gev}\right),$
(17)

so that the ansatz of $r_{0.002}>0.001$, implies $\Lambda\geq 5.9\times
10^{15}\ \mathrm{GeV}$. Figure 3 shows bounds on
$N\left(k_{\mathrm{LIGO}}\right)$ for the range $r=[0.001,0.1]$. In this case
we see that primordial perturbations corresponding to LIGO sensitivity exit
the horizon during inflation within a few e-folds of the end of inflation and
the onset of reheating. This is precisely the region where the assumption of
slow roll breaks down, and we expect parameters such as the expansion rate $H$
and the slow roll parameter $\epsilon$ to be varying relatively rapidly. In
this case, we no longer expect power-law behavior in the primordial tensor
spectrum, and there will be a corresponding systematic uncertainty in the
amplitude of tensors at direct detection frequencies. Figure 4 shows
$N\left(k_{\mathrm{DECIGO}}\right)$ for the planned DECIGO peak sensitivity
range $\nu=0.01-1.0\ \mathrm{Hz}$ [Kawamura  _et al._(2020)Kawamura _et al._],
or $k_{\mathrm{DECIGO}}=6.47\times 10^{13}\ \mathrm{Mpc}^{-1}$ for $\nu=0.1\
\mathrm{Hz}$.

Figure 1: The number of e-folds at horizon exit $N(k_{\mathrm{LIGO}})$ plotted
as a function of inflationary energy scale $\Lambda$, for frequencies $\nu=10\
\mathrm{Hz}$, $\nu=100\ \mathrm{Hz}$, and $\nu=1000\ \mathrm{Hz}$. Red shows
the upper bound (Eq. 15), and blue shows the lower bound (Eq. 16). Regions
where $N\left(k\right)<0$ indicate no production of gravitational waves at
those frequencies. Figure 2: The number of e-folds at horizon exit
$N(k_{\mathrm{DECIGO}})$ plotted as a function of inflationary energy scale
$\Lambda$, for frequencies $\nu=10\ \mathrm{Hz}$, $\nu=100\ \mathrm{Hz}$, and
$\nu=1000\ \mathrm{Hz}$. Red shows the upper bound (Eq. 15), and blue shows
the lower bound (Eq. 16). Regions where $N\left(k\right)<0$ indicate no
production of gravitational waves at those frequencies. Figure 3: The number
of e-folds at horizon exit $N(k_{\mathrm{LIGO}})$ plotted as a function of
tensor/scalar ratio $r$, for frequencies $\nu=10\ \mathrm{Hz}$, $\nu=100\
\mathrm{Hz}$, and $\nu=1000\ \mathrm{Hz}$. Red shows the upper bound (Eq. 15),
and blue shows the lower bound (Eq. 16).  Figure 4: The number of e-folds at
horizon exit $N(k_{\mathrm{DECIGO}})$ plotted as a function of tensor/scalar
ratio $r$, for frequencies $\nu=0.01\ \mathrm{Hz}$, $\nu=0.1\ \mathrm{Hz}$,
and $\nu=1.0\ \mathrm{Hz}$. Red shows the upper bound (Eq. 15), and blue shows
the lower bound (Eq. 16).

In the next section, I use the inflationary flow formalism to show that
extrapolating an exact power-law spectrum to direct detection scales from CMB
scales results in a systematic overestimate of the tensor amplitude at direct
detection scales by as much as two orders of magnitude, relative to realistic
single-field inflation models.

##  2 Flow Analysis

The inflationary flow formalism [Hoffman and Turner(2001), Kinney(2002)] is a
convenient method of generating large ensembles of single-field inflation
models consistent with a particular set of observational constraints [Easther
and Kinney(2003)]. The equations of motion for the spacetime and scalar field
are given by:

$\displaystyle
H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3M_{\mathrm{P}}^{2}}\left[\frac{1}{2}\dot{\phi}^{2}+V\left(\phi\right)\right],$
(18) $\displaystyle\ddot{\phi}+3H\dot{\phi}+V^{\prime}\left(\phi\right)=0.$
(19)

If the field evolution is monotonic in time, we can write the scale factor
$a\left(\phi\right)$ and Hubble parameter $H\left(\phi\right)$ as functions of
the field $\phi$ rather than time. Equations (18) can then be re-written
exactly in the Hamilton-Jacobi form

$\displaystyle V\left(\phi\right)$ $\displaystyle=$ $\displaystyle
3M_{\mathrm{P}}^{2}H^{2}\left(\phi\right)\left[1-\frac{2M_{\mathrm{P}}^{2}}{3}\left(\frac{H^{\prime}\left(\phi\right)}{H\left(\phi\right)}\right)^{2}\right],$
(20) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-2M_{\mathrm{P}}^{2}H^{\prime}\left(\phi\right).$ (21)

The first slow roll parameter $\epsilon$ is related to the equation of state
by

$\epsilon=\frac{3}{2}\left(1+w\right)=-\frac{a}{H}\frac{dH}{da}.$ (22)

Inflation then occurs for $w<-1/3$, or $\epsilon<1$. During inflation, the
scale factor increases quasi-exponentially, with the Hubble parameter
$H\simeq{\rm const.}$ and

$a\propto\exp\left[\int{Hdt}\right]\equiv e^{-N}.$ (23)

Here $N$ is the number of e-folds before the end of inflation, which is
related to the field $\phi$ by

$\displaystyle dN\equiv-Hdt$ $\displaystyle=$
$\displaystyle-\frac{da(\phi)}{a(\phi)}$ (24) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}M_{\mathrm{P}}}\frac{d\phi}{\sqrt{\epsilon(\phi)}}.$
(25)

We can then write the parameter $\epsilon$ as

$\epsilon=\frac{1}{H}\frac{dH}{dN}=2M_{\mathrm{P}}^{2}\left(\frac{H^{\prime}\left(\phi\right)}{H\left(\phi\right)}\right)^{2}.$
(26)

We define an infinite hierarchy of Hubble slow roll parameters [Copeland  _et
al._(1993)Copeland, Kolb, Liddle, and Lidsey, Liddle  _et al._(1994)Liddle,
Parsons, and Barrow] by taking successive derivatives of the Hubble parameter
$H$ with respect to the field $\phi$:

$\displaystyle\epsilon$ $\displaystyle\equiv$
$\displaystyle{2M_{P}^{2}}\left(\frac{H^{\prime}(\phi)}{H(\phi)}\right)^{2},$
(27) $\displaystyle\eta$ $\displaystyle\equiv$
$\displaystyle{2M_{P}^{2}}\frac{H^{\prime\prime}(\phi)}{H(\phi)},$ (28)
$\displaystyle{}^{2}\lambda_{H}$ $\displaystyle\equiv$
$\displaystyle{4M_{P}^{4}}\frac{H^{\prime}(\phi)H^{\prime\prime\prime}(\phi)}{H^{2}(\phi)},$
(29) $\displaystyle\vdots$ (30) $\displaystyle{}^{\ell}\lambda_{H}$
$\displaystyle\equiv$
$\displaystyle{\left(2M_{P}^{2}\right)^{\ell}}\frac{H^{\prime}(\phi)^{\left(\ell-1\right)}}{H(\phi)^{\ell}}\frac{d^{\left(\ell+1\right)}H(\phi)}{d\phi^{\left(\ell+1\right)}},$
(31)

Using Eq. (24) to take derivatives of $\epsilon$ with respect to $N$, we can
generate an infinite set of differential equations relating the parameters:

$\displaystyle\frac{d\epsilon}{dN}$ $\displaystyle=$ $\displaystyle
2\epsilon\left(\eta-\epsilon\right),$ (32) $\displaystyle\frac{d\eta}{dN}$
$\displaystyle=$ $\displaystyle-\epsilon\eta+{}^{2}\lambda_{H},$ (33)
$\displaystyle\vdots$ (34) $\displaystyle\frac{d{}^{\ell}\lambda_{H}}{dN}$
$\displaystyle=$
$\displaystyle\left[(\ell-1)\eta-\ell\epsilon\right]{}^{\ell}\lambda_{H}+{}^{(\ell+1)}\lambda_{H},$
(35)

This hierarchy of flow equations completely specifies the evolution of the
spacetime [Kinney(2002)]: once the parameters $\epsilon\left(N\right)$,
$\eta\left(N\right)$ and so forth are known, the cosmological dynamics and
scalar field potential are fixed via the Hamilton-Jacobi Equations (20). In
practice, the hierarchy of equations must be truncated at finite order, which
I take to be $\ell\leq 6$. Since the system of equations is first-order,
specifying a point in the parameter space is sufficient to fully fix the
solution. (Note that solutions to the truncated system of equations are also
exact solutions to the infinite system.)

To test the validity of a power-law extrapolation from CMB to direct detection
scales, I perform a Monte Carlo evaluation of the flow equations (32),
collecting 500 realizations consistent with Planck+BICEP/Keck 95% confidence
bounds on the scalar spectral index $n_{S}=[0.9569,0.9733]$,
$r_{k=0.002}<0.056$ , with the scalar power spectrum normalized at the pivot
scale to $A_{S}=2.090524\times 10^{-9}$ [Akrami  _et al._(2020)Akrami _et
al._]. I take $N(k=0.002)$ varying between 50 and 60, to ensure that the modes
at short wavelength are redshifted fully into the infrared limit, and make the
additional ansatz of selecting only realizations for which $r_{0.002}\geq
0.001$, consistent with the expected sensitivity of near-future CMB
polarization measurements such as CMB-S4 [Carlstrom  _et al._(2019)Carlstrom,
Abazajian, Addison, Adshead, Ahmed, Allen, Alonso, Alvarez, Anderson, Arnold,
Baccigalupi, Bailey, Barkats, Barron, Barry, Bartlett, Basu Thakur, Battaglia,
Baxter, Bean, Bebek, Bender, Benson, Berger, Bhimani, Bischoff, Bleem,
Bocquet, Boddy, Bonato, Bond, Borrill, Bouchet, Brown, Bryan, Burkhart, Buza,
Byrum, Calabrese, Calafut, Caldwell, Carlstrom, Carron, Cecil, Challinor,
Chang, Chinone, Cho, Cooray, Crawford, Crites, Cukierman, Cyr-Racine, de Haan,
de Zotti, Delabrouille, Demarteau, Devlin, Di Valentino, Dobbs, Duff,
Duivenvoorden, Dvorkin, Edwards, Eimer, Errard, Essinger-Hileman, Fabbian,
Feng, Ferraro, Filippini, Flauger, Flaugher, Fraisse, Frolov, Galitzki, Galli,
Ganga, Gerbino, Gilchriese, Gluscevic, Green, Grin, Grohs, Gualtieri, Guarino,
Gudmundsson, Habib, Haller, Halpern, Halverson, Hanany, Harrington, Hasegawa,
Hasselfield, Hazumi, Heitmann, Henderson, Henning, Hill, Hložek, Holder,
Holzapfel, Hubmayr, Huffenberger, Huffer, Hui, Irwin, Johnson, Johnstone,
Jones, Karkare, Katayama, Kerby, Kernovsky, Keskitalo, Kisner, Knox, Kosowsky,
Kovac, Kovetz, Kuhlmann, Kuo, Kurita, Kusaka, Lahteenmaki, Lawrence, Lee,
Lewis, Li, Linder, Loverde, Lowitz, Madhavacheril, Mantz, Matsuda, Mauskopf,
McMahon, Meerburg, Melin, Meyers, Millea, Mohr, Moncelsi, Mroczkowski,
Mukherjee, Munchmeyer, Nagai, Nagy, Namikawa, Nati, Natoli, Negrello,
Newburgh, Niemack, Nishino, Nordby, Novosad, O’Connor, Obied, Padin, Pandey,
Partridge, Pierpaoli, Pogosian, Pryke, Puglisi, Racine, Raghunathan, Rahlin,
Rajagopalan, Raveri, Reichanadter, Reichardt, Remazeilles, Rocha, Roe, Roy,
Ruhl, Salatino, Saliwanchik, Schaan, Schillaci, Schmittfull, Scott, Sehgal,
Shandera, Sheehy, Sherwin, Shirokoff, Simon, Slosar, Somerville, Staggs,
Stark, Stompor, Story, Stoughton, Suzuki, Tajima, Teply, Thompson, Timbie,
Tomasi, Treu, Tristram, Tucker, Umilta, van Engelen, Vieira, Vieregg,
Vogelsberger, Wang, Watson, White, Whitehorn, Wollack, Wu, Xu, Yasini, Yeck,
Yoon, Young, and Zonca].

Comparison of primordial tensor modes to direct detection bounds on the energy
density in gravitational waves requires application of a linear transfer
function $T(k)$ to the primordial spectrum, taking into account cosmological
evolution after wave modes becomes subhorizon, which is done for an ensemble
of inflationary potentials, for example, in Ref. [Caligiuri  _et
al._(2015)Caligiuri, Kosowsky, Kinney, and Seto]. Since our goal here is to
compare a power-law extrapolation to realistic single-field inflation models,
this step is not necessary: it is sufficient to compare the amplitudes of the
primordial spectra directly at direct detection scales, since linearity
guarantees that any systematic error will be retained after processing by the
transfer function. Both the background evolution and the tensor power spectrum
are calculated numerically, without using the slow roll approximation. The
result is shown in Fig. 6 for LIGO/Virgo, and Fig. 7 for DECIGO. Realistic
inflation models lie as much as two orders of magnitude below the expectation
from extrapolation of a power-law from CMB scales, because the Hubble
parameter typically decreases rapidly near the end of inflation. This is shown
for an example case in Fig. 5. The exact single-field solution differs
strongly from the power-law extrapolation at large $k$.

Figure 5: The exact tensor power spectrum (solid black line) vs. the power-law
extrapolation (dotted red line) for an example inflationary potential. The
power-law form associated with slow roll is a good approximation at small $k$,
but the exact form deviates strongly from power-law near the end of inflation,
at large $k$.

Figure 8 shows the primordial tensor amplitude
$P_{T}\left(k_{\mathrm{LIGO}}\right)$ plotted versus the tensor spectral index
at the CMB pivot scale, $n_{T}\left(k_{0.002}\right)$. The corresponding
result for DECIGO is plotted in Fig. 9. Because the tensor/scalar ratio $r$
and the spectral index $n_{T}$ are related by the consistency relation (6), we
expect the tensor amplitude $P_{T}$ and the spectral index $n_{T}$ to be
highly correlated, which is in fact the case for the power-law extrapolation.
This correlation, however, completely disappears for realistic inflationary
potentials. This means that, for all practical purposes, the tensor amplitude
at direct detection scales gives no information about the tensor spectral
index at CMB scales.

Figure 6: The primordial tensor amplitude
$P_{T}\left(k_{\mathrm{LIGO}}\right)$ for an exact slow roll solution vs. a
power-law extrapolation from the CMB pivot scale $k=0.002\ \mathrm{Mpc}^{-1}$
to the LIGO/Virgo scale $k_{\mathrm{LIGO}}=6.47\times 10^{16}\
\mathrm{Mpc}^{-1}$. The crosses show the extrapolated power-law, which forms a
line of unit slope. The points show the calculated values for the exact slow
roll potentials, which lie as much as two orders of magnitude below. Figure 7:
The primordial tensor amplitude $P_{T}\left(k_{\mathrm{DECIGO}}\right)$ for an
exact slow roll solution vs. a power-law extrapolation from the CMB pivot
scale $k=0.002\ \mathrm{Mpc}^{-1}$ to the DECIGO scale
$k_{\mathrm{DECIGO}}=6.47\times 10^{13}\ \mathrm{Mpc}^{-1}$. The crosses show
the extrapolated power-law, which forms a line of unit slope. The points show
the calculated values for the exact slow roll potentials, which lie as much as
two orders of magnitude below. Figure 8: The primordial tensor amplitude
$P_{T}\left(k_{\mathrm{LIGO}}\right)$ at LIGO/Virgo scales plotted vs. the
tensor spectral index at the CMB pivot scale $k=0.002\ \mathrm{Mpc}^{-1}$. The
red crosses show the values for a power-law extrapolation of the tensor power
spectrum, which are highly correlated due to the single-field consistency
condition. The black points show the values for the exact slow roll
calculation, which are uncorrelated with $n_{T}$ measured at the CMB pivot
scale. Figure 9: The primordial tensor amplitude
$P_{T}\left(k_{\mathrm{DECIGO}}\right)$ at DECIGO scales plotted vs. the
tensor spectral index at the CMB pivot scale $k=0.002\ \mathrm{Mpc}^{-1}$. The
red crosses show the values for a power-law extrapolation of the tensor power
spectrum, which are highly correlated due to the single-field consistency
condition. The black points show the values for the exact slow roll
calculation, which are uncorrelated with $n_{T}$ measured at the CMB pivot
scale.

##  3 Conclusions

Primordial tensor perturbations at superhorizon scales are a unique prediction
of cosmological inflation [Geshnizjani and Kinney(2015)]. The single-field
consistency condition (6) relating the tensor amplitude to the slope of the
power spectrum is a key target for observational tests of inflation. CMB
bounds combined with high-frequency gravitational wave constraints from direct
detection bounds such as the LIGO/Virgo and DECIGO observatories in principle
provide a large lever arm with which to constrain the slope of the primordial
spectrum. Current bounds are complicated by the fact that no primordial
gravitational waves have yet been detected, and it is known to be somewhat
difficult to measure the slope of a function without having measured its value
at any point. In fact, for low enough inflationary energy scale, no primordial
tensor modes at all are produced at direct detection scales. Any constraint
must therefore be model-dependent, for example by assuming a lower bound on
the tensor amplitude at CMB scales by ansatz. Inflation predicts an
approximately power-law spectrum of primordial perturbations over at least a
few orders of magnitude in scale, and it is typically assumed that this power-
law behavior extends to frequencies accessible by direct detection
experiments.

In this paper, I examine the consistency of power-law extrapolation of the
inflationary tensor spectrum from CMB scales to direct detection scales, and
find that such extrapolation overestimates the direct detection signal by up
to two orders of magnitude relative to an ensemble of realistic single-field
inflation models with $r_{0.002}>0.001$ (Figs. 6, 7). This is because high-
frequency modes are produced very near the end of inflation, when the Hubble
parameter is rapidly decreasing, and the tensor amplitude is correspondingly
suppressed (Fig. 5). In addition, there is no correlation between $n_{T}$ at
CMB scales and the tensor amplitude at LIGO/Virgo scales (Fig. 8), and only
weak correlation at DECIGO scales (Fig. 9). This means that any constraint on
$n_{T}$ at CMB scales, and therefore the consistency relation, is highly
model-dependent. Conversely, a detection of primordial tensors at both CMB and
(for example) DECIGO scales would place a strong constraint the form of the
inflationary potential $V\left(\phi\right)$ [Caligiuri  _et
al._(2015)Caligiuri, Kosowsky, Kinney, and Seto]. Until then, the only well-
supported conclusion we can make about the spectral index of the primordial
tensor power spectrum is: we have no idea.

## Acknowledgements

I thank Alessandro Melchiorri, Sunny Vagnozzi, and Eleonora Di Valentino for
helpful discussions. The inflationary flow code was written by Brian A. Powell
and WHK.

###### Acknowledgements.

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