to follow Good and Walker [45], and to introduce states $\phi_k$ diagonalising the $T$ matrix (which e.g. in the proton case describes different $p \to N^*$, $N_a^* \to N_b^*$ transitions). Such eigenstates only undergo elastic scattering. Since there are no off-diagonal transitions,

$$ \langle \phi_i | T | \phi_k \rangle = 0 \quad \text{for } i \neq k, \qquad (20.20) $$

a state $k$ cannot diffractively dissociate into a state $j \neq k$. Working in terms of the Good-Walker eigenstates $\phi_i$, we have a simple one-channel eikonal for each state. We denote the orthogonal matrix which diagonalizes $T$ by $a$, so that

$$ T = aFa^T \quad \text{with} \quad \langle \phi_i | F | \phi_k \rangle = F_k \delta_{ik}, \qquad (20.21) $$

where $F_k$ is the probability amplitude of the hadronic process proceeding via the diffractive eigenstate $\phi_k$.

Now consider the diffractive dissociation of an incoming state $|h\rangle$. We can write

$$ |h\rangle = \sum_k a_{hk} |\phi_k\rangle. \qquad (20.22) $$

The elastic scattering amplitude satisfies

$$ \langle h | T | h \rangle = \sum_k |a_{hk}|^2 F_k = \langle F \rangle, \qquad (20.23) $$

where $F_k \equiv \langle \phi_k | F | \phi_k \rangle$ and where the brackets of $\langle F \rangle$ mean that we take the average of $F$ over the initial probability distribution of diffractive eigenstates. After the diffractive scattering described by $T_{fh}$, the final state $|f\rangle$ will, in general, be a different superposition of eigenstates from that of $|h\rangle$, which was shown in Eq. (20.22). Neglecting the real parts, for the cross sections at a given impact parameter $b$, we have

$$ \begin{aligned} \frac{d\sigma_{\text{tot}}}{d^2b} &= 2 \operatorname{Im}\langle h | T | h \rangle = 2 \sum_k |a_{hk}|^2 \operatorname{Im}F_k = 2\langle\operatorname{Im}F\rangle \\ \frac{d\sigma_{\text{el}}}{d^2b} &= |\langle h | T | h \rangle|^2 = \left| \sum_k |a_{hk}|^2 F_k \right|^2 = \langle |F\rangle|^2 \\ \frac{d\sigma_{\text{el} + \text{SD}}}{d^2b} &= \sum_k |\langle \phi_k | T | h \rangle|^2 = \sum_k |a_{hk}|^2 |F_k^2| = \langle |F^2| \rangle. \end{aligned} \qquad (20.24) $$

It follows that the cross section for the single diffractive dissociation of a proton,

$$ \frac{d\sigma_{\text{SD}}}{d^2b} = \langle |F^2| \rangle - \langle |F| \rangle^2, \qquad (20.25) $$

is given by the statistical dispersion in the absorption probabilities of the diffractive eigenstates. Here the average is taken over the components $k$ of the incoming proton which dissociates. If the averages are taken over the components of both of the incoming particles, then Eq. (20.25) is the sum of the cross sections for single and double dissociation, see Fig. 20.3.

Note that if all the components $\phi_k$ of the incoming proton $|h\rangle$ were absorbed equally, then the diffracted superposition would be proportional to the incident one and the probability of the inelastic diffraction would be zero. Thus if, at very high energies, the amplitudes $F_k$ at small impact parameters are equal to the black disk limit, $F_k = i$, then diffractive production will be equal to zero in this impact parameter domain, and so will only occur in the peripheral $b$ region where the edge of the disk becomes not completely black. Hence the impact parameter structure of diffractive dissociation and elastic scattering is drastically different in the presence of absorptive s-channel unitarity effects (see the $G_{inel}$ term in Eq. (20.9)). Under the assumption that amplitudes $F_k$ at high energies cannot exceed the black disk limit, $\operatorname{Im} F_k \le 1$, equations 20.24 lead to the following bound

$$ \frac{d\sigma_{\text{el} + \text{SD}_1 + \text{SD}_2 + \text{DD}}}{d^2b} \le \frac{1}{2} \frac{d\sigma_{\text{tot}}}{d^2b}. \qquad (20.26) $$

known as the Pumplin bound [46]<fnref n="5" />.

### 20.3.3 Diffraction from the t-channel viewpoint

The t-channel approach is based on the Regge model (see Section 20.2), where high-energy diffractive processes are mediated by the exchange of a Pomeron (P). In the case of the elastic pp-scattering amplitude in the eikonal model (see Eq. (20.13)), the opacity corresponding to the exchange of one Pomeron is

$$ \Omega(s, b) = \int \frac{d^2 q_t}{4\pi^2} \Omega(s, q_t) e^{i\vec{q}_t \cdot \vec{b}} \qquad (20.27) $$

with

$$ \Omega(s, q_t) = \frac{1}{s} T'_{\text{el}} = -i\eta_P(t)g_N(t)g_N(t) \left(\frac{s}{s_0}\right)^{\alpha_P(t)-1}, \qquad (20.28) $$

where $T'_{\text{el}}$ is the two-particle s-channel irreducible elastic amplitude, cf. Eq. (20.5), and $g_N(t)$ is the proton-Pomeron coupling.

If we assume an exponential t-dependence of the coupling, $g_N(t) = g_N(0) \exp(B_0 t)$, and neglect the Pomeron phase, then the opacity is

$$ \Omega(s, q_t) = g_N(0)g_N(0) \left(\frac{s}{s_0}\right)^{\alpha_P(0)-1} e^{Bt}, \qquad (20.29) $$

with the t-slope given by

$$ B = 2B_0 + \alpha'_{\text{P}} \ln \left( \frac{s}{s_0} \right). \qquad (20.30) $$

At high energies the opacity has a Gaussian form in the b-space:

$$ \Omega(s, b) = \frac{g_N^2(0)}{4\pi B} \left(\frac{s}{s_0}\right)^{\alpha_P(0)-1} e^{-b^2/4B}. \qquad (20.31) $$

In terms of opacity the effective radius of interaction increases at high energies as $\sqrt{\alpha'_{\text{P}} \ln(s/s_0)}$. This means that with energy increasing the differential cross section becomes steeper (the so called *shrinkage* of the diffractive peak).

If we were to take for the Pomeron the DL parametrisation [25, 26], that is to keep just the first, $T(b) = \Omega(b)/2$, term in the elastic amplitude (Eq. (20.13)) then, at LHC energies, the Gaussian would exceed the black disk limit at small $b$. However, the eikonal unitarization reduces the power growth of the one-Pomeron exchange cross section. Thus, in Eq. (20.31) $\Omega(s, b) \propto (s/s_0)^{\alpha_P-1}$ gives an amplitude $\operatorname{Im} T_{\text{el}}(s, b) = 1 - e^{-\Omega/2} < 1$. Hence the total cross section is limited by the size of the effective interaction area $\sigma_{\text{tot}} < 2\pi R^2$, where the interaction radius $R$ can be estimated from Eq. (20.31) as the value of $b$ where $\operatorname{Re}\Omega(b)$ becomes $\sim 1$.

For the parameterization of Eq. (20.31) the corresponding radius grows at very large energies as

$$ b^2 = R^2 = 4B \ln \left[ \frac{g_N^2(0)}{4\pi B} \left( \frac{s}{s_0} \right)^{\alpha_P(0)-1} \right] \approx 4\Delta\alpha'_{\text{P}} \ln^2(s/s_0). \qquad (20.32) $$

That is for $\Delta = 0.1$ and $\alpha'_{\text{P}}=0.25 \text{ GeV}^{-2}$ we may expect that the cross section increases as

$$ \sigma_{\text{tot}} = 2\pi R^2 \approx c \cdot \ln^2 s, \qquad (20.33) $$

with $c = 8\pi\Delta\alpha'_{\text{P}} = 0.24 \text{ mb}$. This value is close to that obtained by the COMPETE parameterization ($c = 0.27 \text{ mb [22, 24]}$) but much smaller than the Froissart-Lukaszuk-Martin (FLM) bound [47–49]. With $c^{\text{FLM}} = \pi/m_{\pi}^2 \approx 60 \text{ mb}$, see Section 20.7,

$$ \sigma_{\text{tot}} \le \frac{\pi}{m_{\pi}^2} \ln^2 \left( \frac{s}{s_0} \right). \qquad (20.34) $$

<footnote n="5">⁵Strictly speaking the proof of the Pumplin bound is justified only for low mass dissociation. When the masses $M_{1,2}$ become so large (say, $M_{1,2}^2 > \sqrt{s\bar{s}_0}$) that the Good-Walker states $|\phi_i\rangle$, corresponding to two incoming protons overlap, we may face double counting. Therefore, the high mass dissociation will be considered in the next Section, in terms of the multi-Pomeron diagram. Here and in what follows $s_0$ is a constant which should be defined for a particular theoretical model or fitted from experiment.</footnote>
The fact that $c=(0.24-0.27)$ mb $\ll c^{\text{FLM}} = 60$ mb demonstrates that even at the LHC we are very far from true high-energy asymptotics $^6$, and the observed growth of the cross section is driven by the interactions at relatively large transverse momenta $k_t \gg m_\pi$ rather than the smallest hadron mass $m_\pi$ in the denominator of Eq. (20.34).

### 20.3.3.1 The t-slope and dip in the elastic cross section

We first start with a relatively small one-Pomeron amplitude and consider the two-Pomeron contribution corresponding to the $\Omega^2$ term in the expansion of the eikonal $1 - \exp(-\Omega/2)$. In this term the momentum transferred, $q_t = \sqrt{|t|}$, is divided between the two Pomerons so that each Pomeron carries about a momentum $q_t/2$. Correspondingly, the $t$ dependence of the whole ‘two-Pomeron’ amplitude will be $\exp(2B(t/4)) = \exp(Bt/2)^7$.

Since the two-Pomeron contribution has an opposite sign in comparison with the one-Pomeron exchange, their interference will result in the appearance of the first diffractive minimum which moves to smaller $|t|$ with energy increasing. Such interference effects are largely responsible for the zero in the imaginary part of the amplitude (with the minimum filled by the real part).

It is worth mentioning that the one-channel eikonal discussed so far is a rather oversimplified approximation. It provides some indications about the behaviour we may expect for the elastic cross section, but clearly it does not give the whole story. Moreover, even within the framework of the one-channel eikonal, the expectation for the elastic slope $t$-dependence could be masked by other effects. Firstly, there is no reason why the $t$-dependence of the proton-Pomeron coupling $g_N(t)$ has to be a pure exponent. Next, there exists a two-pion singularity at $t = 4m_\pi^2$ (close to the physical region) in the Pomeron trajectory which generates some curvature in the behaviour of $d\sigma_{el}/dt$ [50–52]. So there may be some compensation between the effects caused by the eikonal (arising from the interference between the different multi-Pomeron contributions), and the curvatures coming from the form of the proton-Pomeron coupling and the two-pion singularity of the Pomeron trajectory. However, an exact compensation looks quite non-trivial and a pure exponential behaviour of $d\sigma_{el}/dt$ looks highly unlikely.

Indeed, the measurements by the TOTEM collaboration at 8 TeV [53] and at 13 TeV [54] clearly demonstrate that the local slope of the elastic pp cross section,

$$B = d[\ln(d\sigma_{el}/dt)]/dt, \qquad (20.35)$$

at $-t \lesssim 0.3 \text{ GeV}^2$ varies with $t$.

### 20.3.3.2 High mass dissociation

Let us turn to inelastic diffractive processes that is, to single and double proton dissociations, $pp \to X+p$ and $pp \to X_1+X_2$, where the $+$ sign denotes the presence of a LRG in the distribution of final state particles. For example, for the diffractive dissociation of a proton into a system of mass $M$, the rapidity gap between the incoming proton and the remaining hadrons is

$$\Delta y = \ln \left( \frac{s}{M^2} \right) = \ln \left( \frac{1}{\xi} \right), \qquad (20.36)$$

where $\xi = 1-x$ and $x$ is the initial momentum fraction (Feynman variable) carried by the outgoing proton. The masses, $M$, of the diffractively excited states, produced in high $\sqrt{s}$ collisions, can be large. To separate dissociation from the common inelastic process, usually the condition $M^2 \ll s$ is imposed.

The simplest multi-Pomeron diagram used to describe the diffractive dissociation is the so-called triple-Pomeron graph, shown at the end of Fig. 20.2.

In the Regge pole model, the cross section for the inclusive single diffractive (SD) dissociation process [55–57] can be written

in the form (see Fig. 20.2)

$$\begin{aligned} \frac{\xi d\sigma_{\text{SD}}}{dt d\xi} &= \frac{M^2 d\sigma_{\text{SD}}}{dt dM^2} \\ &= \frac{g_{3\mathbb{P}}(t) g_N(0) g_N^2(t)}{16\pi^2} \left(\frac{s}{M^2}\right)^{2\alpha_{\mathbb{P}}(t)-2} \left(\frac{M^2}{s_0}\right)^{\alpha_{\mathbb{P}}(0)-1}, \end{aligned} \qquad (20.37)$$

where $g_{3\mathbb{P}}(t)$ is the triple-Pomeron coupling. The value of the coupling $g_{3\mathbb{P}}$ is usually obtained from a triple-Regge analysis of lower energy data (see e.g. [34]).

In an analogous way the cross section for double dissociation reads

$$\begin{aligned} \frac{\xi_1 \xi_2 d\sigma_{DD}}{dt d\xi_1 \xi_2} &= \frac{M_1^2 M_2^2 d\sigma_{DD}}{dt dM_1^2 dM_2^2} \\ &= \frac{g_{3\mathbb{P}}^2(t) g_N^2(0)}{16\pi^3} \left(\frac{ss_0}{M_1^2 M_2^2}\right)^{2\alpha_{\mathbb{P}}(t)-2} \left(\frac{M_1^2 M_2^2}{s_0^2}\right)^{\alpha_{\mathbb{P}}(0)-1}, \end{aligned} \qquad (20.38)$$

where $t$ is the momentum squared transferred through the LRG. As discussed in Section 20.5, from a microscopic point of view the

<figure><img src="image_112.png" /><figcaption>**Figure 20.4:** Pomeron exchange with schematic diagrams for the enhanced and semi-enhanced exchanges.</figcaption></figure>

Pomeron exchange is described by a set of ladder-type diagrams (see [58–60]), which can lead to a rescattering of the intermediate partons (produced inside this ladder during the evolution), see Fig. 20.4. The left plot shows the Pomeron exchange complemented with the rescattering of partons 1 and 2 and the scattering of a parton 3 on the target. In terms of multi-Pomeron exchanges this corresponds to the diagram on the right hand side, where the Pomeron exchange is shown by the double line of a corresponding colour. The blue one is called “enhanced” (its contribution is integrated over the rapidities of both upper and lower vertices, i.e. of partons 1 and 2). The loop formed by the Pomerons shown in red is called “semi-enhanced” (it is integrated over the rapidity of one intermediate parton).

While the rescattering of the incoming hadron (proton) is already embedded in the eikonal formula (Eq. (20.13)), the rescattering of the intermediate partons in RFT is accounted for by the so-called enhanced diagrams $^8$ with multi-Pomeron vertices, $g_m^n$, which couple $m$ to $n$ Pomerons. It is quite a challenging task to resum all the enhanced diagrams, however this was successfully performed within the framework of the QGSJET Monte Carlo [61]. An elegant approach to sum up all enhanced diagrams in the case when each extra effective Pomeron contribution is very large was proposed in [62], assuming the analyticity of the $g_m^n$ vertices in $n$ and $m$ in the right half of the complex n- and m-planes. The resulting amplitude becomes a black disk.

The simplest triple-Pomeron vertex $g_2^1 = g_{3\mathbb{P}}$ produces the first multi-Pomeron graph considered above (see the end of Fig. 20.2). However, numerically the multi-Pomeron vertices are relatively small. Note also that the value of $g_{3\mathbb{P}}$, determined from the fit to

<footnote>$^6$As usual, we assume $s_0 = 1 \text{ GeV}^2$, but the qualitative conclusion does not depend on any realistic choice of $s_0$.</footnote>

<footnote>$^7$The two-Pomeron contribution has a factor of two smaller $t$-slope, and in terms of the impact parameter, the $\Omega^2(b)$ term is concentrated in the domain of a smaller radius. In such a simplified picture, the impact parameters corresponding to an exchange of $n$ Pomerons will rapidly decrease with $n$ increasing.</footnote>

<footnote>$^8$This contribution is enhanced due to the large parton multiplicity.</footnote>
experimental data (e.g. [63]), is actually an effective vertex with coupling

$$g_{\text{eff}} = g_{3\mathbb{P}} \langle S^2 \rangle, \qquad (20.39)$$

which already includes the survival factor $S^2(b)$, see Eq. (20.15).

Since the opacity $\Omega$ increases with energy, at large $\Omega$ the number of multiple interactions grows as $N \propto \Omega$, leading to a smaller $S^2$. An explicit analysis [64] accounting for the survival effects gives a coupling $g_{3\mathbb{P}}$ about a factor of 3 larger than $g_{\text{eff}}$, namely $g_{3\mathbb{P}} \simeq 0.2g_N$.

Recall that the Pomeron exchange simultaneously describes both the elastic scattering amplitude, $T_{\text{el}}$, and the multiparticle production cross section, $G_{\text{inel}}$. The discontinuity (disc $T_{\text{el}}$) of the ladder diagram corresponds to the production of secondary particles, practically homogeneously distributed over the whole available rapidity interval covered by the Pomeron, as illustrated by the right-hand diagram in Fig. 20.5.

<figure><img src="image_113.png" /><figcaption>Figure 20.5: Cut Pomeron contribution to the inelastic cross section.</figcaption></figure>

For the one-Pomeron case this discontinuity is called the “cut Pomeron”. Correspondingly each multi-Pomeron diagram describes a series of different processes. Cutting $k$ Pomerons in the diagram with $n$ Pomerons we get the inelastic interaction with the multiplicity (density of secondaries) $k$ times larger than that, $N_0$, produced by one cut Pomeron, $dN/dy = k \cdot N_0$. The remaining $n-k$ (elastic) Pomerons account for the absorptive corrections to the subprocess with $k$ cut Pomerons. Indeed, the contribution of the diagram with $n$ Pomerons includes also the processes with larger, $(k+i) \cdot N_0$ multiplicities (cut Pomerons), where $(i=1, 2, ..., n-k)$. Absorptive corrections, described by the remaining elastic Pomerons, play a role of the survival factor $S^2$ for the process with the fixed particle density $k \cdot N_0$. They ensure probability conservation (the sum of the probabilities of all possible different channels is equal to one) and restore unitarity. Note that the multi-Pomeron diagrams represent all possible interactions between partons from the protons and partons from the Pomerons. In the case of Monte Carlo generators, the non-enhanced multi-Pomeron contributions are included in terms of the multiple parton interaction (MPI) option, see [13,65] and Section 7.2 in [2]. However, as a rule, this option accounts mainly for the multiple interactions between the partons from the protons (incoming hadrons). The energy-momentum sharing between the various inelastic rescattering processes (including the cut and un-cut Pomerons) was performed at the amplitude level within the EPOS Monte Carlo [66].

#### 20.3.3.3 AGK cutting rules

The relation between the cross sections of subprocesses with a different number of cut Pomerons within a given diagram with $n$ Pomerons) is given by the AGK (Abramovsky-Gribov-Kancheli [67]) cutting rules. These rules include also the cut between the Pomerons with $k=0$ which corresponds to the contribution of the particular diagram to the elastic cross section. By applying these rules, it is possible to show the self-consistency of the approach, which was lacking in the pure Regge-pole model.

Consider a diagram where the elastic scattering amplitude is mediated by an exchange of $n$ Pomerons. The AGK cutting rules specify the coefficients $c_n^k$ arising when $k$ of these Pomerons are cut. Recall that the Pomeron cut discontinuities give the

corresponding inelastic contributions to $\sigma_{\text{tot}}$. The terms with $k=0$ correspond to the diffractive cutting of the diagram (that is, the cut is between the Pomeron exchanges, and not through the Pomerons themselves), while the terms with $k=1, 2, ...$ describe the processes with $k$ cut Pomerons. The coefficients $c_n^k = \sigma_n^k / |\sigma_{\text{tot}}^{(n)}|$ are $^9$

$$c_n^{k=0} = (-1)^n (2^{n-1} - 1), \qquad c_n^{k \neq 0} = \frac{(-2)^{n-1} \frac{(-1)^{k-1} n!}{k!(n-k)!}}{(20.40)}$$

where $\sigma_{\text{tot}}^{(n)}$ denotes the contribution of the $n$-Pomeron diagram to the total cross section. Note the alternating sign of $\sigma_{\text{tot}}^{(n)}$ expressed as $(-1)^{n-1}$.

<figure><img src="image_114.png" /><figcaption>Figure 20.6: Two-Pomeron exchange diagram as a sum of different AGK cuts shown by the dashed lines.</figcaption></figure>

For the two-Pomeron exchange, $n=2$, the coefficients are $+1, -4$, or $+2$ according to whether $k=0, 1$ or $2$ Pomerons are cut, respectively. As shown in Fig. 20.6, the amplitude of the two-Pomeron exchange corresponds to a sum of three processes: i) inelastic interaction with particle density twice that caused by one Pomeron (see Fig. 20.6(c)) which enters with the coefficient '2', ii) shadowing (absorptive) correction to the one-Pomeron exchange contribution, which corresponds to events with a single Pomeron density (only one Pomeron is cut), see Fig. 20.6 (b), which enters with a factor '-4', and iii) diffractive elastic scattering or proton dissociation (when different components of the proton wave function correspond to different interaction cross sections), caused by the distortion of the incoming plane wave, see Fig. 20.6(a).

Note that the inclusive cross section is not affected by the multi-Pomeron contribution: $2 \times (2) + 1 \times (-4) = 0$. This is a general property of the AGK rules valid for any number of Pomerons $n$. Thus in order to calculate the inclusive single-particle cross section, it is sufficient to consider just the one-Pomeron exchange diagram.

Let us emphasize that the AGK rules provide a framework to consistently work with multi-Pomeron diagrams, that is, with the Regge cuts, accounting for their contributions to different processes (elastic scattering and diffractive dissociation, inelastic events with different densities, $dN/dy$, of secondaries, etc.).

Measurements of diffractive dissociation cross sections have been made in a wide range of pre-LHC energies, see e.g. [68–73]. At the LHC, cross sections of events with a LRG were measured by the ATLAS, CMS and ALICE collaborations at 7 and 8 TeV,

<footnote n="9">⁹In their complete form the AGK cutting rules were implemented in the QGSJET Monte Carlo [61].</footnote>
see [74–77]. ATLAS [78] and CMS and TOTEM [79] presented first measurements of SD cross sections at 8 TeV with a tagged forward proton. While ATLAS measured inclusive SD cross section, CMS and TOTEM studied SD dijet production. Note that in [78] the measured slope $B = 7.65 \pm 0.34 \text{ GeV}^{-2}$ of the inclusive SD cross section as well as the differential distributions $\frac{\xi d\sigma_{\text{SD}}}{d\text{d}\xi}$ for $0.0001 \le \xi \le 0.025$ are (within the experimental uncertainties) in a good agreement with the theoretical expectations [29,80]. Moreover a relatively small (in comparison with the $d\sigma_{\text{el}}/dt$) slope $B$ indicates that the size of the triple-Pomeron vertex is much smaller than the proton size.

### 20.3.4 Central Diffractive processes

Processes $pp \to p + X + p$, where an object $X$, produced in the central rapidity region, is separated from the outgoing protons by a LRG on each side, are called Central Exclusive Production (CEP). They are described by the double Pomeron exchange (DPE) diagrams. When the mass of the central system, $M_X$, is large and the interaction in the $M_X$ region can be described by Pomeron exchange, the corresponding cross section reads

$$ \frac{\xi_1 \xi_2 d\sigma^{\text{CEP}}}{d\xi_1 dt_1 d\xi_2 dt_2} = \frac{g_N^2(t_1) g_N^2(t_2)}{(16\pi^2)^2} \left(\frac{1}{\xi_1}\right)^{2\alpha_F(t_1)-2} \left(\frac{1}{\xi_2}\right)^{2\alpha_F(t_2)-2} \\ \times g_{3\text{P}}^2(0) \left(\frac{M_X^2}{s_0}\right)^{\alpha_F(0)-1} . \qquad (20.41) $$

If the mass $M_X$ is not too large or for the cases (such as exclusive Higgs boson or dijet production) where the mass $M_X$ is comparable with the corresponding hard scale, the last factor $g_{3\text{P}}^2(0)(M_X^2/s_0)^{\alpha_F(0)-1}$ should be replaced by the corresponding ‘Pomeron-Pomeron cross section’, see for instance [81, 82].

Note that equations (20.37), (20.38) and (20.41) are written in a simplified way without accounting for absorptive corrections. That is, the cross sections in equations (20.37) (20.38) and (20.41) should be multiplied by the gap survival factor $S^2$ (see Eq. (20.15)).

Since the QCD Pomeron is built mainly from gluons it is natural to search for glueballs in double Pomeron exchange processes, and in particular, in CEP.

Resonance production in the Pomeron-Pomeron fusion was extensively studied at the CERN ISR at $\sqrt{s}$ from 22 GeV to 63 GeV (see for reviews [82–84]) and, after the ISR closure in 1983, in fixed target experiments at the CERN SPS [85] and E690 at the Tevatron [86, 87]. Glueballs were actively searched for and the properties of the $f_0$ and $f_2$ production studied in detail using multiparticle spectrometers, such as the Omega facility at the CERN SPS experiments (WA76, WA91 and WA102), see for a review [85].

An important property of CEP processes, which can be expected from matching with the perturbative QCD LO (leading order) calculation, is the $J_z = 0$ dominance. Perturbatively, for the CEP of a heavy object, the leading contribution comes from a configuration with the projection of this object spin onto the beam axis $J_z = 0$ [81]. Note that the CEP cross section is suppressed at large $M_X$ by a strong bremsstrahlung off the incoming gluons (from the Pomeron) which would violate the ‘exclusivity’. The small probability of not having such radiation is described by the Sudakov suppression factor, $T_{\text{Sud}}$, [88], see [81] for details.

### 20.3.5 Diffractive parton distributions

Selecting in Deep Inelastic Scattering (DIS) events with a LRG (see e.g. [89,90]) or detecting the leading proton (see Section V.C. in the review [91]) we can study the parton (quark and gluon) distributions of the Pomeron $^{10}$. In other words, such events can be treated as DIS on the Pomeron target with the incoming Pomeron flux given by

$$ f_{\mathcal{P}}(x_{\mathcal{P}}) = \int dt \frac{g_N^2(t)}{16\pi^2} x_{\mathcal{P}}^{2(1-\alpha_{\mathcal{P}}(t))}, \qquad (20.42) $$

where the proton momentum fraction transferred through the Pomeron $x_{\mathcal{P}} = \xi = M^2/s$.

These Pomeron PDFs were extracted from the HERA measurements of ep scattering with leading protons or a LRG and can be used to describe the inclusive production of high $E_T$ dijets or another hard process based on the collinear factorization theorem in the same way as that in non-diffractive collisions (see [91]). The inclusive measurements of these PDFs are described in [92–94], with the combined H1 and ZEUS data using tagged protons analyzed in [95]. The impact of diffractive jet measurements is addressed e.g. in [96] and the measured charm contribution is presented in [97, 98]. As far as the parton distributions are known, we can calculate the corresponding inelastic cross section of the Pomeron-proton interaction using one of the ‘general purpose’ Monte Carlo generators (see e.g. [13]), multiply it by the Pomeron flux and compare the obtained result with the Regge formula in Eq. (20.37). This approach provides another way to evaluate the triple-Pomeron vertex $g_{3\text{P}}$. The corresponding analysis was performed in [99] and leads to practically the same (within the error bars) value of $g_{3\text{P}} = 0.2g_N(0)$.

It is worth mentioning that in DIS at large $Q^2$ we are dealing with small-size objects and the rescattering effects are small. Therefore, the survival factor $S^2 \simeq 1$ and does not affect the results.

## 20.4 Experimental data on diffraction at high energies

### 20.4.1 Total and elastic cross sections

The elastic scattering of protons is a process with a special and rather simple experimental signature: the central detector is empty while the incoming protons after the collisions are detected in the dedicated forward proton detectors (FPD) placed far from the interaction point (IP). Elastic scattering data are taken in special runs in order to be able to reach different ranges of $t$-values and thanks to the very large value of the cross section the data can be collected with a relatively low instantaneous luminosity and hence a negligible pile-up.$^{11}$

These special runs usually have very few proton bunches and differ in the $t$ range covered, which is governed roughly by the relation $t_{\min} \propto d^2/\beta^*$. Here $d$ is the distance, expressed in multiples of the beam size at the detector, from the centre of the LHC beam and $\beta^*$ is defined as the distance from the IP to the point where the transverse area of the beam is twice as wide as that at the IP (see Section 31 in [1]). Note that if we work at large $\beta^*$, the incoming protons have very small angular divergence leading to small average transverse momentum, which allows us to measure very small $|t|$ values. The lowest $|t|$ values measured so far at the LHC are $4 \times 10^{-4} \text{ GeV}^2$ (ALFA) and $6 \times 10^{-4} \text{ GeV}^2$ (TOTEM) reached with the 8 TeV LHC beam configured with $\beta^*=1$ km optics. The largest $t$ values of about $4 \text{ GeV}^2$ were measured by TOTEM at 8 and 13 TeV with $\beta^*=90$ m thanks to special triggers. Other $\beta^*$ values used in special runs are 3.5 m, 11 m and 2.5 km.

There are four ways to determine the $\sigma_{\text{tot}}$ value:

1. **Elastic and Inelastic.** This method does not require the optical theorem and hence no extrapolation of $d\sigma_{\text{el}}/dt$ to $t=0$ and no $\rho$ (defined below Eq. (20.17)) but rather the luminosity and measuring rates $N_{\text{el}}$ (elastic) and $N_{\text{inel}}$ (in-elastic). The total cross section is then simply:

$$ \sigma_{\text{tot}} = \frac{1}{\mathcal{L}}(N_{\text{el}} + N_{\text{inel}}). \qquad (20.43) $$

Of course, both $N_{\text{el}}$ and $N_{\text{inel}}$ should be corrected for the detector acceptance and efficiency. This is especially important for $N_{\text{inel}}$ since the detectors never cover the whole rapidity region (i.e. the whole $4\pi$).

2. **Elastic only.** This approach necessitates measuring $d\sigma_{\text{el}}/dt$ and using the optical theorem with a known value of $\rho$. As explained in Section 20.2, the optical theorem states that

<footnote>$^{10}$see also Section 18.5 in [1]</footnote>

<footnote>$^{11}$The pile-up is formed by additional pp collisions which typically produce low-$p_T$ particles. These may affect the signal sample and worsen various reconstruction and identification efficiencies.</footnote>
<figure><img src="image_115.png" /><figcaption>Figure 20.7: Overview of elastic ($\sigma_{el}$), inelastic ($\sigma_{inel}$) and total ($\sigma_{tot}$) cross section data for pp and $p\bar{p}$ collisions as a function of $\sqrt{s}$. The continuous black lines (lower for pp, upper for $p\bar{p}$) represent the best fits of the total cross section data by the COMPETE collaboration [22]. The dashed line is a fit of the elastic cross section data. The dashed-dotted lines refer to the inelastic cross section and are obtained from the difference between the continuous and dashed lines. Figure from Ref. [100].</figcaption></figure>

$\sigma_{tot} \propto \text{Im}[T_{el}(t \to 0)]$, see Eq. (20.3). Since in practice it is not possible to measure down to $t = 0$, we need to extrapolate. To minimize the model dependence when extrapolating, it is vital to measure down to as low $|t|$ values as possible (i.e. high $\beta^*$). This method requires an independent luminosity measurement. Once the luminosity is known, $d\sigma_{el}/dt$ can be normalized and used to extract $\sigma_{tot}$ using the formula:

$$ \sigma_{tot}^2 = \frac{16\pi}{1 + \rho^2} \left. \frac{d\sigma_{el}}{dt} \right|_{t \to 0} . \qquad (20.44) $$

3. **Coulomb normalization.** Similarly to the previous method, this approach relies on the elastic observables only and requires a measurement of the elastic cross section at very low values of $|t|$, where it is sensitive to the theoretically well known Coulomb QED contribution $4\pi\alpha^2_{QED}/t^2$. The normalization of $d\sigma_{el}/dt$ is then determined by fitting the experimental data at very low $|t|$ using a formula including the Coulomb amplitude and its interference with the strongly interacting (the so-called nuclear) term. This method has been successfully used by UA4/2 [101] and TOTEM [102].

4. **Luminosity-independent.** This method does not rely on the knowledge of luminosity but rather on the knowledge of $N_{el}$ and $N_{inel}$ and on the optical theorem: combining equations (20.43) and (20.44) with $\frac{d\sigma_{el}}{dt} = \frac{1}{L} \frac{dN_{el}}{dt}$ we get

$$ \sigma_{tot} = \frac{16\pi}{1 + \rho^2} \left. \frac{dN_{el}/dt}{N_{el} + N_{inel}} \right|_{t=0}, \qquad (20.45) $$

where $dN_{el}/dt|_{t=0}$ corresponds to the extrapolation to $t = 0$ of the nuclear term only. By independently and simultaneously measuring $N_{el}$ and $N_{inel}$, and applying the optical theorem, we can also determine the luminosity.

The TOTEM [100, 103–105] and ATLAS [106, 107] collaborations at CERN have covered an energy range from $\sqrt{s}=2.76$ TeV to 13 TeV. A compilation of high energy total pp and $p\bar{p}$ cross section measurements is shown in Fig. 20.7 (for discussion of the pre-LHC elastic scattering data see review [108]).

Despite some tension between the Tevatron CDF [109] and E811 [110] data 12 and to a lesser extent between the TOTEM [104, 105] and ATLAS [106, 107] measurements, the data clearly indicate that in the Tevatron – LHC energy interval the total cross section starts to grow faster than the power-law parametrization [26] describing the data below the Tevatron energy. In particular, while the DL fit [26] predicts $\sigma_{tot} = 90.7$ mb at $\sqrt{s} = 7$ TeV, the TOTEM experiment observes $98.6 \pm 2.2$ mb [104].

A compilation of the high-energy data on the elastic slope is shown in Fig. 20.8. It is clearly seen that in the TeV energy range the slope increases with $\sqrt{s}$ more rapidly than the logarithmic behaviour expected in the case of one-Pomeron exchange, see Eq. (20.30). Such an acceleration of the $t$-slope derivative, $dB/d\ln s$, is a clear manifestation of the increasing role of the multi-Pomeron exchanges, where asymptotically the slope should rise as $\ln^2 s$, see [117]. Finally, Fig. 20.9 illustrates the energy dependence of the differential elastic pp cross section. As expected (see Section 20.3.3.1), the diffractive dip moves to smaller $|t|$ with increasing energy.

### 20.4.2 Diffractive vector meson production

The exclusive production of vector mesons was studied in detail at HERA (see for a review [91]). It is well described within the ‘dipole model’ (see for review and references [120]), where the incoming photon first fluctuates into a quark-antiquark, which then interacts with the target proton and, finally, with the probability given by the overlap integral between the vector meson wave function and the outgoing $q\bar{q}$-pair, the vector meson is produced. The crucial quantity is the value of cross section, $\sigma(q\bar{q} - p)$, of elastic scattering of the $q\bar{q}$-pair on the proton. The energy behaviour of $\sigma(q\bar{q} - p)$ is driven by the intercept, $\alpha_{eff}(0)$, of the effective Pomeron 13 (rightmost singularity in the $j$-plane), while

<footnote>12The CDF 1.8 TeV point [109] is 2.8 $\sigma$ higher than the corresponding E811 result [110].</footnote>

<footnote>13Effective Pomeron means that this is not an original pole in the $j$-plane, but it includes the corrections (renormalizations) caused by the enhanced diagrams (see e.g. [121]).</footnote>
<figure><img src="image_116.png" /><figcaption>**Figure 20.8:** The diffractive slope *B* for pp and pπ⁻ elastic scattering as a function of √s. The experimental uncertainties represent the quadratic sum of statistical and systematic uncertainties. The dashed line is a result of a linear fit to data at √s < 3 TeV. The data points come from [103–107, 111–113]. Figure from Ref. [100].</figcaption></figure>

<figure><img src="image_117.png" /><figcaption>**Figure 20.9:** The t-dependence of the pp elastic cross section for collision energies $\sqrt{s} = 2.76$ TeV [103], 7 TeV [104, 114], 8 TeV [53, 115, 116] and 13 TeV [54, 102]. The experimental uncertainties represent the quadratic sum of statistical and systematic uncertainties. Figure from Ref. [116].</figcaption></figure>

the value of the cross section depends on the quark separation, *r*, in the transverse plane, $\sigma(q\bar{q} - p) \propto \alpha_s^2 \langle r^2 \rangle$ [43, 44]. Thus different processes with the same $\langle r^2 \rangle$ are driven by the same $\sigma(q\bar{q} - p)$ cross section.

In the DIS case this separation in turn is controlled by the photon virtuality, $Q^2$, and the quark mass, $m_q$: $\langle r^2 \rangle \simeq 1/(z(1-z)Q^2 + m_q^2)$ ($z$ is the photon momentum fraction carried by the quark). Indeed, the cross section of the $\rho$ meson diffractive production in DIS at $Q^2 = M_{J/\psi}^2$ is close (up to the difference in the quark electric charges) to that for the $J/\psi$ photoproduction, see Fig. 20.10 (Left).

The production cross section depends non-trivially on W, the

energy of the $\gamma^*p$ center of mass system. It increases with W as $W^n$, where $n = 0.2$ for $\rho, \omega$ and $\phi$ (light quark)-mesons but $n = 0.8$ for $J/\psi$. Note that in the $J/\psi$ case the energy dependence is close to that of the BFKL (Balitsky-Fadin-Kuraev-Lipatov) Pomeron [59, 122, 123], that is, the singularity calculated within the leading (and next-to-leading) approximation in perturbative QCD. But at lower scales the absorptive (multi-Pomeron) corrections tame the growth which leads to smaller values of $n$ ([124–127]), see Fig. 20.10 (Right).

A similar situation reveals in the dependence of $\alpha_{\text{eff}}$ on $Q^2$, as can be seen in Fig. 47 of [91]. At a large scale $\mu^2 = (Q^2 + M_V^2)/4$ the value of $\alpha_{\text{eff}} \simeq 1.3$ is close to the prediction for the QCD
<figure><img src="image_118.png" /><figcaption>Figure 20.10: (Left) The $\rho, \omega, \phi$ and J/ψ elastic production cross sections as a function of the scale $\mu^2 = (Q^2 + M_V^2)/4$. For readability of the figure, the J/ψ cross sections are multiplied by a factor 2. Figure from Ref. [118]. (Right) Compilation of photoproduction cross section measurements as a function of the $\gamma p$ centre-of-mass energy, W. The total cross section and various vector meson production cross sections are included, with the approximate power law dependences $\sigma \propto W^\delta$ indicated for each process. Figure from Ref. [119].</figcaption></figure>

Pomeron, while for a smaller scale, the absorptive corrections described by the multi-Pomeron diagrams start to reduce the cross section, and $\alpha_{eff}$ decreases.

## 20.5 Pomeron in QCD

All features described in the previous Sections were based on first principles, such as analyticity (based on causality), unitarity, crossing symmetry, etc. Since QCD theory satisfies all these principles it should reveal a corresponding “Regge” behaviour. Indeed, within perturbative QCD there is a Pomeron: an even-signature singularity in the j-plane with vacuum quantum numbers. While in the old Regge theory the Regge trajectories and their couplings were phenomenological numbers fitted from experiment, perturbative QCD allows one to calculate the positions of the singularities and the corresponding couplings with $O(\alpha_s)$ and even with $O(\alpha_s^2)$ accuracy [59, 122, 123, 128–131].

In terms of Feynman diagrams, the QCD Pomeron may be viewed as a sum of multi-particle ladders built by the exchange of two t-channel (reggeized $^{14}$) gluons, see the left-hand side of Fig. 20.5.

The sum of ladder diagrams of the type of Fig. 20.5 is the simplest multiparticle structure which reproduces the power-like $s^\alpha$ behaviour of the Pomeron pole. In other words it corresponds to a sum of completely inelastic $2 \to n$ processes, that is, to the last term $G_{inel} = 1 - \exp(-\Omega)$ in the unitarity equation (20.9). This set of diagrams was resummed in the limit of a small QCD coupling, $\alpha_s \ll 1$, but large energy, such that $\alpha_s \ln(s/s_0) \sim O(1)$ [59]. The summation results in the rightmost singularity at $j = 1 + \omega_0 > 1$. After accounting for the next-to-leading logarithmic (NLL) corrections, the position of the singularity (Pomeron intercept) corresponds to $\omega_0 = 0.25–0.3$ depending only weakly on the scale [122, 123, 132–136], whose value is characterized by the transverse momentum, $k_t$, of gluons in the ladder.

It was demonstrated (see e.g. [137]) that the resummation of

the $(\alpha_s \ln(1/x))^n$ terms based on the QCD Pomeron results essentially improves the description of low-$x$ inclusive HERA data within the framework of the NNLO DGLAP evolution.

At this stage the singularity is the cut in the j-plane. However we have to account for the boundary conditions at relatively small $k_t$. Imposing a reasonable boundary, we arrive at a series of Regge poles in the interval from $j = 1$ to $j = 1 + \omega_0$ instead of the cut [134]. Note that the first (corresponding to the rightmost pole in the j-plane, i.e. to the pole with the largest Re $j$) eigenfunction consists of gluons with relatively small $k_t$, while for the next poles the $k_t$ increases. DIS inclusive $\gamma^*p$ cross sections were fitted in [135] using the QCD based approach in which Pomeron is represented by series of Regge poles obtained within the perturbative QCD BFKL approach. It was concluded that the first pole has a small coupling to the proton. It is possible that this small value of the coupling to the proton is related to the fact that the enhanced multi-Pomeron diagrams (i.e. the rescattering of intermediate partons) were neglected in the fit. The main effect of this enhanced contribution is the “renormalization” of the intercept which diminishes the effective value of $\omega_0$. Besides this, the enhanced diagrams provide a saturation by reducing the rise of the parton densities in the $(b, k_t, y)$-space (see e.g. [138, 139]).

Note that perturbative QCD allows us to understand why the values of the phenomenological multi-Pomeron vertices and the shift, $\omega_0$, of the intercept, are small (due to $\alpha_s \ll 1$ and some numerical factors such as $N_c$ and $\pi$). Indeed, at the lowest $\alpha_s$ orders we get for the $\omega_0$ value and the simplest multi-Pomeron vertices (see e.g. [59, 139, 140]):

$$ \omega_0 \propto \frac{N_c \alpha_s}{\pi}, \quad g_{3P} \propto \frac{N_c \alpha_s^2}{(N_c^2 - 1)\pi^2} \quad \text{and} \quad g_2^2 \propto \frac{N_c \alpha_s}{(N_c^2 - 1)^2}, \qquad (20.46) $$

where $g_2^2$ is the coupling corresponding to the transition of 2 into 2 Pomerons.

### 20.5.1 BFKL evolution in the 'dipole' representation

It was shown in [141–144] that the LO BFKL Pomeron equation [59] can be written in terms of the evolution of the dipole

<footnote>$^{14}$That is, the virtual loop corrections to the one-gluon exchanges are included. These corrections are important in order to provide infrared stability of the results.</footnote>
density, $N(x_d, y_d; y)$, in rapidity $y$ (here $x_d$ and $y_d$ are the transverse coordinates of two $t$-channel gluons which form the colour singlet dipole). Indeed, after the emission of a new gluon at point $z_d$, the initial colour dipole with coordinates $(x_d, y_d)$ turns into a pair of dipoles $(x_d, z_d)$ and $(z_d, y_d)$. This can be considered as a development of a ‘dipole cascade’. Moreover in this formalism it is easy to include the non-linear absorptive corrections (last term in the square brackets in Eq. (20.47)), which accounts for the rescattering of the intermediate partons (gluons) on the target proton. The corresponding contribution is described by the so-called “fan” diagrams and these are the most important corrections to the linear DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) evolution [145] in the case of DIS at not large scales but at very small momentum fraction [139].

The resulting non-linear evolution (Balitsky-Kovchegov equation [146–148]) reads

$$ \frac{d}{dy} N(x_d, y_d; y) = \frac{\alpha_s N_c}{2\pi^2} \int d^2 z_d \frac{(x_d - y_d)^2}{(x_d - z_d)^2 (y_d - z_d)^2} \times [N(x_d, z_d; y) + N(y_d, z_d; y) - N(x_d, y_d; y) - N(x_d, z_d; y)N(y_d, z_d; y)] . \quad (20.47) $$

For a small density $N$ the last term in the square brackets can be neglected, and the first three terms in Eq. (20.47) reproduce the conventional BFKL equation in the coordinate representation. However, for large $N \to 1$ the right-hand side of Eq. (20.47) vanishes and we reach the saturation $N = 1$. It is worth mentioning that, as shown in [149], in terms of ‘dipole’ formalism, with the triple-Pomeron vertex generated by the ‘one dipole to two dipoles’ transition, it is possible to relate the Good-Walker approach to high mass diffraction with the triple-Pomeron diagram.

## 20.5.2 Distribution of secondaries: theory versus experiment

As already discussed, in terms of Feynman diagrams the cut Pomeron can be viewed as a set of ladder diagrams corresponding to a sum of completely inelastic $2 \to n$ processes, that is, to the last term $G_{inel} = 1 - \exp(-\Omega)$ in the unitarity equation (20.9). Here $n > 2$ means the production of additional $(n-2)$ gluons which, after hadronization, form minijets.<fnref n="15" /> Therefore, in the final state driven by one Pomeron, we expect to observe gluon minijets with a flat rapidity distribution in the central (plateau) rapidity region. This would correspond to a flat pseudorapidity distribution of produced particles if they were massless. A typical pseudorapidity distribution of charged particles in inclusive events (up to $|\eta| = 7$) is shown in Fig. 20.11 (left) [150] (see also Fig. 53.1 in [1]). The central part ($|\eta| < 2.5$) was measured by CMS, while the forward region was covered by TOTEM. The dip observed at $\eta = 0$ is explained by the presence of massive particles (the Jacobian $J(p_T, m, \eta) = p_T/E \to p_T/\sqrt{p_T^2 + m^2}$ at $\eta = 0$). A photon energy spectrum is shown in Fig. 20.11 (right) [151], measured by LHCf inclusively and in events with a diffraction topology, i.e. no charged particles with $p_T > 100$ MeV and $|\eta| < 2.5$ observed by ATLAS. As expected in diffractive events the energy flow decreases with $E_\gamma$ more slowly than that in the inclusive case.

The energy dependence of the particle density $dN_{ch}/d\eta$ at $\eta = 0$ is shown in Fig. 20.12 (left). Neglecting absorptive corrections given by the enhanced diagrams (which mainly change (‘renormalize’) the effective Pomeron intercept $\alpha_{eff}(0) = 1 + \Delta$ [121]), we conclude that according to the AGK rules the plateau height $d\sigma/d\eta \propto s^\Delta$ is driven just by the one-Pomeron exchange with effective $\Delta \sim 0.2$ (see Section 20.3.3.3). That is, the density of secondaries observed in the inclusive process increases with increasing energy faster than the total cross section, whose growth is tamed by the multi-Pomeron diagrams. Indeed, as is seen from Fig. 20.12 (left), in the interval of collider energies $dN_{ch}/d\eta = (1/\sigma_{inel})d\sigma/d\eta \propto s^{0.115}$ (i.e. $d\sigma/d\eta \propto s^{0.215}$), while $\sigma_{inel} \propto s^{0.1}$.

Contrary to the ‘old’ Regge theory where it was assumed (based on the experimental data existing in the 1950s and 1960s) that all transverse momenta are limited, in QCD the $k_t$ distributions of jets (charged particles) have a long $k_t$ tail ($d\sigma/dk_t^2 \propto \alpha_s^2(k_t^2)/k_t^4$ at large $k_t$ and very large energy $s \gg k_t^2$). An example of the $p_T$ distribution of charged secondaries is shown in Fig. 20.12 (right).

Note that the mean transverse momentum of secondaries, produced via jet fragmentation, slowly increases with collision energy, see Fig. 20.13 (right). This is caused by the stronger absorption (at larger $\sqrt{s}$) of the gluons with a smaller $k_t$ ($\sigma^{abs} \propto 1/k_t^2$). The growth of $\langle p_T \rangle$ with multiplicity (see Fig. 20.13 (left)) can be explained by the fact that events with larger $N_{ch}$ correspond to a smaller impact parameter, $b$, where the absorption of a low $k_t$ component is stronger and, next, larger multiplicity can be originated by the events with jets/minijets with higher $p_T$. Since the mean $p_T$ of secondaries grows with $\sqrt{s}$, the increase with $\sqrt{s}$ of transverse energy flow is a bit faster than that of particle density.

The model [162] based on a modification of the classic RFT allows one to trace the smooth transition from the pure perturbative, large $k_t$, region into the soft domain. A strong absorption of the low $k_t$ partons plays a crucial role here since it produces an effective infrared cutoff, $k_{sat}$, and provides the possibility of extending the parton approach, used for ‘hard’ processes, to also describe high-energy soft and semihard interactions. This approach combines a description of soft physics and diffraction with jet physics in a coherent self-consistent way.

Another way is to include the soft and hard components independently [37, 66, 163, 164]. In this approach the soft part is described in terms of RFT with the phenomenological “soft” Pomeron pole while the hard part is calculated in terms of the parton model for minijet production with the energy dependent cutoff $k_t > k_0(s)$. A combined description of soft and hard processes in hadronic collisions is reached within the QGSJET Monte Carlo model (e.g. [61]) in the framework of the so-called “semi-hard Pomeron” approach (see e.g. [165]).

In [166] a model was constructed, which incorporated the attractive features of the two successful theoretical approaches to high energy QCD: BFKL Pomeron calculus [59,60] and the Colour Glass Condensate/saturation [167].

### 20.5.2.1 Correlations

All LHC experiments routinely measure tracks with $p_T > p_{min}$, where $p_{min}$ can vary in different studies. Typically, $p_{min} = 200$ MeV, where tracking reconstruction efficiencies are larger than 70%. In order to identify particle species, each experiment has sophisticated identification procedures usually based on the ionization energy loss, $dE/dx$, or other techniques, with different regions of applicability for different particle species. Thanks to usually relatively large cross sections of soft QCD processes, most of the results below come from event samples with very low or negligible pile-up.

Following the notation in [168], symmetrized inclusive particle number densities for $q$ points at $y_1, ..., y_q$ (where $y_i$ represents the 4-momentum of the $i$th particle), $\rho_q(y_1, ..., y_q)$, are related to the inclusive differential cross section by

$$ \frac{1}{\sigma_{inel}} d\sigma = \rho_1(y)dy, \quad \frac{1}{\sigma_{inel}} d^2\sigma = \rho_2(y_1, y_2)dy_1dy_2 \quad \text{etc.} \quad (20.48) $$

By integrating we get

$$ \int \rho_1(y)dy = \langle n \rangle, \quad \iiint \rho_2(y_1, y_2)dy_1dy_2 = \langle n(n-1) \rangle \quad \text{etc.,} \quad (20.49) $$

where the angular brackets denote averaging over the event sample and $n$ is the particle multiplicity.

Since the inclusive $q$-particle densities in general contain trivial contributions from lower-order densities, it is convenient to consider quantities $C_q$ which vanish when one of their arguments becomes statistically independent of (uncorrelated with) the others. These quantities $C_q$, called correlation functions (or cumulant

<footnote n="15">¹⁵Minijets result from hadronization of partons emitted from the cut QCD Pomeron. Typically these are groups of hadrons with comparatively low overall $E_T \lesssim 5–10$ GeV.</footnote>
<figure><img src="image_119.png" /><figcaption>Figure 20.11: (Left) Charged-particle pseudorapidity distribution for inclusive events measured by CMS and TOTEM [150]. The error bars represent the statistical and uncorrelated systematic uncertainties between neighboring bins, while the shaded areas denote the combined statistical and full systematic uncertainties. The coloured lines indicate model predictions. (Right) Photon energy spectrum measured by LHCf at $|\eta| > 10.94$. The filled circles show the inclusive photon spectrum measured by LHCf [152] and filled squares the spectrum for $N_{ch} = 0$ events where no charged particles with $p_T > 100$ MeV and $|\eta| < 2.5$ are observed by ATLAS [153]. The coloured lines indicate model predictions. The error bars correspond to the statistical uncertainties and the shaded areas denote the combined statistical and systematic uncertainties. Figure from Ref. [151].</figcaption></figure>

<figure><img src="image_120.png" /><figcaption>Figure 20.12: (Left) Energy dependence of the charged particle density $dN_{ch}/d\eta$ at $\eta \approx 0$ for pp and $p\bar{p}$ collisions. Shown are measurements performed with different Non-SD event selections from UA1 [154], UA5 [155], CDF [156,157], ALICE [158] and CMS [159]. The dashed line is a power-law fit to the data. Figure from Ref. [150]. (Right) Differential cross section of charged particles with $|\eta| < 0.8$ in inelastic pp collisions at $\sqrt{s} = 0.9, 2.76$ and 7 TeV as a function of $p_T$. Only statistical uncertainties are shown. Figure from Ref. [160].</figcaption></figure>

functions), are defined as:

$$ 
\begin{aligned}
C_2(1,2) &= \rho_2(1,2) - \rho_1(1)\rho_1(2), & C_3(1,2,3) \\
&= \rho_3(1,2,3) - \sum_{(3)} \rho_1(1)\rho_2(2,3) + 2\rho_1(1)\rho_1(2)\rho_1(3), \\
C_4(1,2,3,4) &= \rho_4(1,2,3,4) - \sum_{(4)} \rho_1(1)\rho_3(1,2,3) \\
&\quad - \sum_{(3)} \rho_2(1,2)\rho_2(3,4) + 2\sum_{(6)} \rho_1(1)\rho_1(2)\rho_2(3,4) \\
&\quad - 6\rho_1(1)\rho_1(2)\rho_1(3)\rho_1(4).
\end{aligned} 
\quad (20.50) 
$$

The 2D two-particle correlation function is defined as

$$ C(\Delta\eta, \Delta\phi) = \frac{\rho_2(\Delta\eta, \Delta\phi)}{\rho_1(\eta_a, \phi_a)\rho_1(\eta_b, \phi_b)}. \quad (20.51) $$

The distribution $\rho_2(\Delta\eta, \Delta\phi)$ is usually interpreted as a conditional probability to observe a particle a at the phase-space point $(\eta_a, \phi_a)$ if a particle b at $(\eta_b, \phi_b)$ is observed as well, and $\Delta\eta = \eta_a - \eta_b$ and $\Delta\phi = \phi_a - \phi_b$. The distributions $\rho_1(\eta_a, \phi_a)$ and $\rho_1(\eta_b, \phi_b)$ are probabilities to observe a single particle at $(\eta_a, \phi_a)$ and $(\eta_b, \phi_b)$, respectively. The denominator of Eq. (20.51) is constructed as a product of two single-particle distributions using an event mixing technique, where each particle in the pair comes from a different event. Experimentally, each reconstructed track is weighted by the inverse of an efficiency factor which accounts
<figure><img src="image_121.png" /><figcaption>**Figure 20.13:** Average $p_T$ of pions, kaons and protons in the range $|\eta| < 1.0$ as a function of (left) track multiplicity at $|\eta| < 2.4$ and (right) of center-of-mass energy where the curves show linear fits using lns. The error bars indicate the uncorrelated combined uncertainties, while the boxes show the uncorrelated systematic uncertainties. Figures from Ref. [161].</figcaption></figure>

for the detector acceptance, the reconstruction and particle identification efficiencies, the contamination by secondary particles and the fraction of misreconstructed tracks.

An example of two-particle correlation functions measured in pp collisions at 7 TeV is shown in Fig. 20.14 for identical-particle pairs (right panel) and for particle-anti-particle pairs (left panel) [169].

We observe two distinct features which can be explained by short-range (in rapidity) correlations: 1) a near-side peak at $\Delta\phi \approx 0$ and 2) an away-side peak or rather a ridge at $\Delta\phi \approx \pi$. The near-side peak is considered to be caused by at least three effects:

* **fragmentation of partons scattered at a hard scale.** These relatively high $p_T$ partons produce showers which after the hadronization form the mini-jets which create a broad structure extending over at least one unit in $\Delta\eta$ and $\Delta\phi$.

* **resonance decays.** The decay of resonances contributes to the near-side peak at $\Delta\eta \sim 0$ and extended in $\Delta\phi$ [170–172], depending on the released kinetic energy of the given resonance. This effect is mostly visible for unlike-sign particle pairs.

* **femtoscopic correlations.** The term “femtoscopic” refers to a length scale of the order of $10^{-15}$ m. These correlations are present at low relative momenta of the particles in a pair (representing a very small phase-space corner, so they are practically invisible in terms of $(\Delta\eta, \Delta\phi)$) and give rise to an enhancement of the correlation function (due to Bose-Einstein quantum statistics for identical bosons) or its suppression (due to Fermi-Dirac quantum statistics for identical fermions). Besides this, at low relative momenta there are correlations caused by Coulomb and/or other final state interactions. The shape of all these effects in $(\Delta\eta, \Delta\phi)$ space depend strongly on the mass of the particle type as well as on the size of the particle-emitting system. The latter is traditionally measured in Bose-Einstein correlation (BEC) analyses and is not part of this review.

The away-side peak originates from energy-momentum conservation which manifests itself by the quark and the anti-quark going back-to-back in $\phi$. In this case the rapidity width of the away-side peak is much larger than the near-side peak since in the original matrix element the quark and the antiquark can be separated by some $\Delta\eta$ interval.

As discussed in Sections 20.3.3.2 and 20.3.3.3, there may be several cut Pomerons in the same event, each giving rise to particle sets which are, in general, independent of each other (except for small Bose-Einstein correlations). This leads to long-range (in rapidity) correlations. Since the density of secondaries, $dN/dy$, is proportional to the number of cut Pomerons, $k$, the probability to observe at least one particle is proportional to $\langle k \rangle$, while the probability to observe simultaneously two particles separated by some (rather large) rapidity interval is proportional to $\langle k^2 \rangle$. Thus the long-range correlations are predicted to be $C_2 = \langle k^2 \rangle / \langle k \rangle^2 - 1 > 0$ which depends weakly on the separation $\Delta\eta$ between the two particles [173,174]. In the case of the pure eikonal approach, neglecting the enhanced diagrams and the conservation law effects in the proton fragmentation region, we expect that these long-range correlations,

$$C_2(\Delta y) = \frac{\sigma_{\text{inel}} d^2 \sigma/dy_1 dy_2}{d\sigma/dy_1 d\sigma/dy_2} - 1 \sim \text{const}, \quad (20.52)$$

do not depend on the rapidity separation, $\Delta y = |y_1 - y_2|$, between the two particles. The contribution of the processes with more cut Pomerons also results in a much wider multiplicity distribution and in a larger density of soft particles coming from the 'underlying event'.

### 20.5.2.2 Color reconnection

In this context, we have to mention also the so-called 'colour reconnection' phenomenon. This is a pure 'soft QCD' effect. The point is that after a number of coloured secondary partons are produced, there are different possibilities to form the colour flow between these partons and to group the partons into colourless clusters. In the process of reconnection, one rearranges the colour flow in such a way as to minimize the size of the clusters. This is especially important when dealing with MPI contributions. The reconnection between the different cut Pomerons diminishes the final multiplicity and can change the form of the $N_{ch}$ distributions (see e.g. Section 43.3.2 of [1] and [2, 13, 177, 178]).

### 20.5.2.3 Double parton scattering

The probability of MPI depends on the spatial distribution of partons in the incoming protons. The effects of MPI are suppressed if the density of partons is low and the partons from the incoming beam particles are separated from each other by a large
<figure><img src="image_122.png" /><figcaption>**Figure 20.14:** Two-particle correlation functions for identical-particle pairs: $\pi^+\pi^+ + \pi^-\pi^-$, $K^+K^+ + K^-K^-$, $pp + \bar{p}\bar{p}$, and $\Lambda\Lambda + \bar{\Lambda}\bar{\Lambda}$ (left panel) and particle-anti-particle pairs: $\pi^+\pi^-$, $K^+K^-$, $p\bar{p}$ and $\Lambda\bar{\Lambda}$ (right panel). Figure from Ref. [169].</figcaption></figure>

interval in transverse coordinate space $\vec{x}_t$. Events in which two hard subprocesses, caused by interactions of two different parton pairs (say, $(a_1b_1)$ and $(a_2b_2)$), take place simultaneously, are called Double Parton Scattering (DPS). The DPS cross section is driven by the ‘double parton distributions’, $D(y_{a_1}, y_{a_2}, ...)$, where $y_{a_1}$ and $y_{a_2}$ are momentum fractions carried by the partons from the proton $a$ and the dots denote all other coordinates. As a rule, experiments study DPS processes at relatively small momentum fractions $y_i$. Here, correlations due to momentum conservation (like $y_{a_1} + y_{a_2} < 1$) are not so important, and with a reasonable accuracy we can assume a factorization

$$D(y_{a_1}, y_{a_2}, ...) \propto F(y_{a_1}) \cdot F(y_{a_2}), \qquad (20.53)$$

where $F(y_{a_i})$ are the single parton distributions. In such a case the DPS cross section takes the form

$$\sigma^{\text{DPS}} = c \cdot \frac{\sigma_{a_1 b_1} \sigma_{a_2 b_2}}{\sigma_{\text{eff}}}, \qquad (20.54)$$

where $\sigma_{a_1b_1}$ and $\sigma_{a_2b_2}$ are cross sections for the two independent hard processes, while $\sigma_{\text{eff}}$ characterizes the mean area occupied by the partons $a_1$ and $b_1$; the constant factor $c = 1/2$ if both hard processes $(a_1b_1)$ and $(a_2b_2)$ are identical, otherwise $c = 1$. Thus the DPS cross section is sensitive to the spatial separations between partons in the proton (see Section 7.2.3 in [2] and [179, 180] for more explanations and reviews).

One problem is that within this approach we *assume* that the partons $a_1$ and $a_2$ are produced by two independent parton show-ers (and similarly for the other incoming proton). On the other hand, there is a probability that from the beginning we start with the evolution of a single shower which further splits into two differ-ent branches. In this case the separation between the two partons (two shower branches) becomes very small — of the order of the inverse scale ($\sim 1/\sqrt{q^2}$) at which the splitting occurs. The exact value of this ‘splitting’ scale $q^2$ depends on the particular kinematics of the DPS process. So, different experiments (with
<figure><img src="image_123.png" /><figcaption>**Figure 20.15:** (Left) The two-particle cumulant, $c_2\{2, |\Delta\eta| > 2\}$, as a function of $\langle N_{ch}(p_T > 0.4 \text{ GeV}) \rangle$ for pp collisions at $\sqrt{s} = 5.02$ and 13 TeV, $pPb$ collisions at $\sqrt{s_{NN}} = 5.02$ TeV and low-multiplicity $PbPb$ collisions at $\sqrt{s_{NN}} = 2.76$ TeV. The data are constructed from particles with $0.3 < p_T < 3.0 \text{ GeV}$. Figure from Ref. [175]. (Right) The $v_2\{2, |\Delta\eta| > 2\}$, $v_2\{4\}$ and $v_2\{6\}$ values as a function of number of charged particles, averaged over $0.3 < p_T < 3.0 \text{ GeV}$ and $|\eta| < 2.4$, in pp collisions at $\sqrt{s} = 13 \text{ TeV}$. Figure from Ref. [176]. The error bars correspond to the statistical uncertainties, while the shaded areas denote the systematic uncertainties.</figcaption></figure>

different kinematical conditions) can give somewhat different values of $\sigma_{eff}$. In general, the value of $\sigma_{eff}$ depends on the following features: a) on the measured process since the spatial ($b_t$) distributions of different incoming partons (light quarks, heavy quarks, gluons) can be different; b) on the splitting scale, $\sqrt{q^2}$, of one parton cascade into two branches. The typically high value of the splitting scale then explains the fact that the experimentally measured values of $\sigma_{eff} \sim 7-25 \text{ mb}$ (see Fig. 4 of [181]) are smaller than $\sigma_{tot}$ or mostly even lower than the proton area $\pi R_p^2 \sim 22-24 \text{ mb}$ (see e.g. [182]); c) on the $p_T$ balance, $k_T$, in the individual hard process (e.g. for two dijet productions $k_T = |\vec{p}_{T1} + \vec{p}_{T2}|$ where $p_{T1}$ and $p_{T2}$ are jet $p_T$'s of the first hard process (similarly for the second hard process). A small value of $k_T$ indicates that there were no splittings or the splitting scale $\sqrt{q^2}$ was small and, therefore, we expect larger $\sigma_{eff}$; d) on the contribution of single parton scatterings misidentified as DPS. For a lower scale of the hard process this contribution is larger (see [183] for more detailed discussion).

#### 20.5.2.4 Final state interactions

The formalism of the RFT does not include ‘final state interactions’<fnref n="16" />. Therefore, besides the correlations considered in the previous Section 20.5.2.1 we have to expect the correlation caused by partons and hadrons rescattering in the final state. These effects are not crucial at lower energies, but become more important at high LHC energies, in particular in heavy-ion collisions where the particle density is large. For example, the final state interactions (FSI) lead to the formation of the collective flow of secondaries (see e.g. [185] for a review), especially in high-multiplicity events. To study the collective flow experimentally, one has to subtract correlations coming from few-particle sources such as resonance decays, mini-jets, multi-jets and BEC (so called “non-flow”). The non-flow can efficiently be suppressed using the sub-event method, that is by studying the azimuthal correlations between particles separated in $\eta$ [186], or subtracted using the multi-particle correlation (or cumulant) techniques.

The cumulant method is based on calculating 2k-particle azimuthal correlations, $corr_n\{2k\}$, and cumulants $c_n\{2k\}$ (where $k = 1, 2, ...$), for nth Fourier harmonics. The $corr_n\{2k\}$ are defined as [187, 188]:

$$ \langle\langle\text{corr}_n\{2\}\rangle\rangle = \langle\langle e^{in(\phi_1 - \phi_2)}\rangle\rangle, \quad \langle\langle\text{corr}_n\{4\}\rangle\rangle = \langle\langle e^{in(\phi_1 + \phi_2 - \phi_3 - \phi_4)}\rangle\rangle, $$

$$ \langle\langle\text{corr}_n\{6\}\rangle\rangle = \langle\langle e^{in(\phi_1+\phi_2+\phi_3-\phi_4-\phi_5-\phi_6)} \rangle\rangle $$

and similarly for higher numbers of correlated particles. The double-brackets $\langle\rangle$ denote averaging first over particles in an event and then over events within a given event class. For every event, the average is taken over all possible combinations of azimuthal angles $\phi_i$ ($l = 1, ..., 2k$) of the 2k particles. The cumulants are then obtained from multi-particle azimuthal correlations after subtracting correlations between $2(k-1)$ particles according to the following formulae [187, 188]:

$$ \begin{aligned} c_n\{2\} &= \langle\langle\text{corr}_n\{2\}\rangle\rangle, \quad c_n\{4\} = \langle\langle\text{corr}_n\{4\}\rangle\rangle - 2\langle\langle\text{corr}_n\{2\}\rangle\rangle^2, \\ c_n\{6\} &= \langle\langle\text{corr}_n\{6\}\rangle\rangle - 9\langle\langle\text{corr}_n\{2\}\rangle\rangle \times \langle\langle\text{corr}_n\{4\}\rangle\rangle + 12\langle\langle\text{corr}_n\{2\}\rangle\rangle^3. \end{aligned} $$

The cumulants for higher particle multiplicities are calculated in [187, 188]. The cumulants then serve to estimate the Fourier harmonics $v_n$ as follows [187]:

$$ v_n\{2\} = \sqrt{c_n\{2\}}, \quad v_n\{4\} = \sqrt[4]{-c_n\{4\}}, \quad v_n\{6\} = \sqrt[6]{c_n\{6\}}/4. $$

Some of the long-range correlation ($|\Delta\eta| > 2$) results obtained on a sample of charged particles with $0.3 < p_T < 3.0 \text{ GeV}$ and $|\eta| < 2.4$ are summarized in Fig. 20.15. The left plot shows the cumulant $c_2$ measured for pp, $pPb$ and $PbPb$ collisions [175], while the right plot shows the elliptical harmonics $v_2$ measured for pp collisions [176], both as functions of multiplicities of charged particles. The two-particle correlations are observed to be strongest and rising with $N_{ch}$ for $PbPb$ collisions, and weakest and rather flat for pp collisions. The elliptical-flow harmonics for 4- and 6-particle correlations show again a rather flat multiplicity dependence (at least for large multiplicities). Within experimental uncertainties, the values of $v_2\{2\}$, $v_2\{4\}$ and $v_2\{6\}$ measured in pp collisions at 13 TeV are consistent with each other. The similarity between $v_2\{4\}$ and $v_2\{6\}$ suggests that some collective effects are occurring in pp collisions at high multiplicity and the observations are similar to those in $PbPb$ collisions, where the $v_2\{4\}$ values were measured to be close to $v_2\{6\}$ but they are both lower than $v_2\{2\}$ (not shown here)).

Another example of long-range correlations is the so-called “ridge effect”. Here not only the ‘back-to-back’ jet correlations are registered, but also an excess of particles going in the same (in the azimuthal plane) direction as the leading (relatively high $p_T$) hadron. Moreover, this excess is seen at the rapidities separated from the leading hadron by a rather large interval (see e.g. [189] for a review).

It is popular to describe such FSI effects within the hydrodynamic model [190], which operates with collective (thermodynamic) variables. In terms of microscopic interactions, the collec-

<footnote n="16">¹⁶In general, final state interactions can be included into the detailed structure of the multi-Pomeron vertices. However these vertices are phenomenological objects which are not well known experimentally.</footnote>
<figure><img src="image_124.png" /><figcaption>**Figure 20.16:** Mean charged particle density (left) and the sum of transverse momenta of secondaries (right) in events with the 'leading' high $p_T$ particle as a function of $p_T^{\text{lead}}$ in the transverse, towards and away azimuthal regions. Secondaries with $p_T > 0.5$ GeV and $|\eta| < 2.5$ are registered. The error bars (mostly hidden by the data markers) represent combined statistical and systematic uncertainties. Figures from [184].</figcaption></figure>

tive flow can be caused by the geometry of a particular collision (the absorption is smaller for the secondaries flying in the direction orthogonal to the impact parameter vector $\vec{b}$ [191, 192]), or by the colour reconnection at the hadronization stage [193], or accounting for the rescattering of secondaries directly, as was done, for example, in the AMPT model [194].

### 20.5.3 The underlying event

Except for the exclusive case, any 'hard' subprocess is accompanied by soft secondaries coming from initial state radiation (ISR), final state radiation (FSR) and multiple parton interaction (MPI), see Subsection 7.2.2 in [2]. These extra particles distort the signal we are looking for. In particular, they affect the isolation criteria applied to photons and charged leptons and the vertex reconstruction efficiency. In general, also the effects of colour reconnection (discussed in Section 20.5.2.2) contribute to the underlying event.

The usual procedure of estimating the amount of underlying event (UE) is to spatially divide tracks in each event according to their azimuthal angle into the Toward region (where the highest $p_T$ jet points), the Away region (opposite to the Toward region) and to two Transverse regions. The standard observables are the average track multiplicity per unit area and the average scalar sum of track $p_T$ per unit area. Figure 20.16 shows the particle density and the sum of $p_T$ for the UE in ATLAS events containing at least one charged particle with $p_T > 0.5$ GeV and $|\eta| < 2.5$ [184].

Note that by construction the largest values of $\langle p_T \rangle$ are observed in the 'Toward' region, while in the 'Away' region we observe a slightly larger density than in the Toward region. These are results of the 'leading' and 'backward' jet fragmentation. In the transverse region, mostly filled by particles from the UE, the particle density and sum of $p_T$ per unit ($\Delta\eta, \Delta\phi$) area practically do not depend on the $p_T^{\text{lead}}$ since these secondaries come from the other cut Pomeron(s), that is, from other 'multiple interactions'. For low $p_T^{\text{lead}} < 2$ GeV the distributions in all three regions are close to each other. These events actually do not contain a 'hard' subprocess. Moreover, for a very small $p_T^{\text{lead}} \to 1$ GeV we start to select soft events with abnormally low $p_T < p_T^{\text{lead}}$ particles. Since only particles with $p_T > 0.5$ GeV are registered, the signal drops fast for $p_T^{\text{lead}} \to 1$ GeV. As a function of collision energy $\sqrt{s}$, the energy flow in the transverse region increases as $\sum p_T \sim s^{0.2}$ (as follows from Fig. 7 (right) in [184]) due to the larger number of MPI collisions and larger $\langle p_T \rangle$ in each collision $^{17}$. As follows from this and other UE-dedicated LHC studies [195], from the comparisons of the data to the models with and without MPI, the necessity of MPI is convincingly demonstrated.

## 20.6 The Odderon

Apart from the even-signature singularity (Pomeron), in QCD with $N_c = 3$ there exists its counterpart, the odd-signature singularity placed at $j \simeq 1$ and formed by three t-channel reggeized gluons connected in colour space by the symmetric $d^{abc}$ tensor of the colour $SU(3)$ group [200, 201]. This object is called the Odderon. The Odderon exchange amplitude has opposite sign for $pp$ and $p\bar{p}$ scatterings. Its intercept is predicted to be very close to $j = 1$ [202–204], while according to perturbative estimates the coupling to the nucleon is rather small [205, 206]. The corresponding amplitude is mainly real and is about 100 times smaller than the imaginary part of the Pomeron exchange amplitude. Calculating the elastic amplitude via the eikonal formula (20.13) we have to replace the opacity $\Omega(b)$ by the sum $\Omega = \Omega_{\text{even}} + \Omega_{\text{odd}}$, where $\Omega_{\text{even}}$ is mainly real and $\Omega_{\text{odd}}$ is imaginary. Note that at $t = 0$ this QCD Odderon does not couple to mesons, and the t-slope of the Odderon amplitude is expected to be smaller than that for the Pomeron; instead of the singularity at $t = 4m_{\pi}^2$ in the Pomeron case, the nearest singularity in the Odderon channel is at $t = 9m_{\pi}^2$, see for instance [204]. Thus, in the impact parameter $b$ space the QCD Odderon occupies an area of a smaller radius, see e.g. [207].

Experimentally an indication in favour of a manifestation of the high energy C-odd amplitude was observed by comparing the elastic $pp$ and $p\bar{p}$ cross sections in the dip region (where the contribution from the C-even amplitude has a minimum) at the CERN-ISR [197], see Fig. 20.17 (left) $^{18}$.

To get a better understanding of the Odderon effects it would be very instructive to have the $d\sigma_{el}/dt$ data for both $pp$ and $p\bar{p}$ reactions at the same but higher energy $\sim 1$ TeV (ideally in the same apparatus) and, in the ideal case, to study the energy dependence.

At the moment we can only compare the $pp$ cross section measured by TOTEM at $\sqrt{s} = 2.76$ TeV [103] with the $d\sigma/dt$ values measured by the D0 collaboration at 1.96 TeV in $p\bar{p}$ collisions [199], see Fig. 20.17 (right).

The situation looks quite intriguing, but needs further investigation. Note that in the TeV energy range the $\omega, \rho$ and $\omega P, \rho P$ exchange contributions, which may be responsible for the difference between the $pp$ and $p\bar{p}$ cross sections in the dip region at the ISR energies, are practically negligible.

Another way to search for the Odderon is to measure the real part of the elastic $pp$ scattering amplitude via the interference with the pure QED one-photon exchange. Since the one-photon

<footnote>$^{17}$Recall the stronger absorption of the low $k_t$ partons.</footnote>

<footnote>$^{18}$Note that a qualitatively similar behaviour to the $p\bar{p}$ ISR data, namely a filling in of the dip in the t-distribution, was observed by the UA4 collaboration at the CERN $Spp\bar{s}$ collider at $\sqrt{s}=546$ and 630 GeV (see [208, 209] and in particular Fig. 2 in [209]) and by the D0 collaboration at the Tevatron at 1.96 TeV [199].</footnote>
<figure><img src="image_125.png" /><figcaption>**Figure 20.17:** Comparison of the t-dependence of the elastic cross sections from pp and $p\bar{p}$ collisions. (Left) Data from the ISR energy of 53 GeV are shown by closed triangles [196] for pp collisions and by open circles [197] and open squares [198] for $p\bar{p}$ collisions. Only t-dependent uncertainties are shown and the systematic scale uncertainty is estimated to be ±30%. Figure from Ref. [197]. (Right) Data from the D0 experiment at 1.96 TeV [199] are compared with data from the TOTEM experiment [103]. The green dashed line indicates the normalization uncertainty of the D0 measurement. Figure from Ref. [103].</figcaption></figure>

<figure><img src="image_126.png" /><figcaption>**Figure 20.18:** The dependence of the $\rho$ parameter on the collision energy. The pp (blue) and $p\bar{p}$ (green) data are taken from [1]. The TOTEM measurements are marked in red. The two points at 13 TeV correspond to two fit cases, discussed in [102], using the same data. The lines represent fits to the data using the COMPETE parameterization [22]. Figure from Ref. [102].</figcaption></figure>

exchange amplitude contribution is sizeable only at very small $|t|$, this way we can study the Odderon at or near to $t=0$. Indeed, the value of the ratio $\rho \equiv \text{Re}T_{\text{el}}/\text{Im}T_{\text{el}}$, obtained by TOTEM at 13 TeV ($\rho = 0.10 \pm 0.01$ [102]), turns out to be smaller than that expected for the pure even-signature amplitude, see Fig. 20.18.

Based on dispersion relations and assuming the C-even contribution only, from the known total cross sections we would rather expect $\rho \simeq 0.13-0.14$. The difference could indicate that the rise of the total cross section at energies above those of the LHC slows down (see the dispersion relation, Eq. (20.19)) or this could be attributed to an Odderon contribution (see e.g. [210]). However, the Odderon exchange amplitude extracted in this analysis has opposite sign to that for the lowest-$\alpha_s$-order QCD Odderon, see e.g. [205, 206, 211, 212]. Besides this, the Odderon contribution to $\rho$, obtained in [210], grows with $\sqrt{s}$ (for $\sqrt{s} > 0.5$ TeV), while in QCD we expect that the Odderon contribution to $\rho$ decreases with energy, since the QCD Odderon intercept is smaller than that of the QCD Pomeron. Note also that recently the ATLAS Collaboration has published new results on elastic pp scattering at 13 TeV using the ALFA proton spectrometer [213]. While the measured value of $\rho = 0.098 \pm 0.011$ appears to be in a good agreement, a smaller total cross-section (with a smaller error bar)

$\sigma_{\text{tot}} = 104.7 \pm 1.1$ mb, is obtained when both are compared with TOTEM results at 13 TeV [102].

The values of $\sigma_{\text{tot}}$ measured by ATLAS at 7, 8, and 13 TeV are found systematically lower than the TOTEM numbers. We note that if we used these ALFA values in dispersion relations, we would obtain a smaller $\rho$ ratio for the C-even amplitude as well. Indeed, as shown in Figure 3 of Ref. [214], a good description of both, the $\sigma_{\text{tot}}$ and $\rho$ measurements, is achieved based on a global fit to the data performed using a damped $\ln^2 s$ amplitude and without an assumption of the Odderon. A good description of the ATLAS-ALFA data ($\chi^2/\text{n.d.f.} = 0.85$, see table 11 of Ref. [213]) of both $\sigma_{\text{tot}}$ and $\rho$ without an assumption of the Odderon is also obtained in the KMR model [215].

It is worth mentioning also that the Odderon contribution is strongly screened by the multi-Pomeron diagrams, which facilitate the falling-off of $\rho$ with energy increasing, see [216, 217]. On the other hand, analyzing the whole ensemble of high energy elastic pp ($p\bar{p}$) low $|t|$ data, a reasonable description can be obtained using the even-signature amplitude only, that is, without the Odderon. In particular, the RR(PL2)qc model/version of the COMPETE parameterization is consistent with the TOTEM 13 TeV data on
$\sigma_{tot}$ and $\rho$ within $1\sigma$ $^{19}$. Another example is the recent analysis in [212] of the low $|t| < 0.1$ GeV$^2$ elastic data. Fitting all the low-$t$ $pp$ and $p\bar{p}$ data in the range of $\sqrt{s}$ between 13 GeV and 13 TeV without Odderon, Donnachie and Landshoff [212] succeeded to describe the TOTEM cross section with less than $1\sigma$ deviation in each $d\sigma_{el}/dt$ point (see Fig. 8 of [212]). Note that in this analysis, they get a larger value of $\rho$ close to 0.14 at 13 TeV.

It was proposed also to search for the Odderon in exclusive C-
even meson (π⁰, η, f₂, η_c, ...) photoproduction (see e.g. [218,219]).
However the expected cross sections are small (e.g. for η_c) and
in each channel there is a large background caused either by
Pomeron-Pomeron fusion (such as CEP of the f₂ meson produc-
tion in pp or pPb collisions) or due to the vector meson radiative
decay (such as ω → π⁰γ for the case of pion) [220]. Up to now,
no definitive Odderon signal in the C-even meson production has
been observed. At the moment there exist only upper limits on
the photoproduction cross sections obtained in the measurements
at HERA at √s ≃ 200 GeV [221–223].

To conclude, let us emphasize that the existence of the C-odd singularity with intercept $\alpha_{\text{odd}}(0) \simeq 1$ is a firm prediction of QCD. At least in the high $k_t$ region there is a well established C-odd three-gluon contribution to the scattering amplitude. However the expected coupling of such an Odderon singularity is numerically very small. Therefore it is quite challenging to observe its manifestation experimentally. Currently it seems to be a bit premature to draw any definite conclusion about an experimental observation of the Odderon signal.

20.7 Asymptotics

The high-energy behaviour of total hadronic cross sections has
been one of the oldest problems of strong interactions over many
decades, beginning from Heisenberg [224]. The most important
bound obtained based on general analytical properties of scat-
tering amplitudes is the FLM bound [47–49]. It states that the
growth of the total hadronic cross section with energy does not
exceed $\ln^2$s, see Eq. (20.34).

Recall that we neglected the photon contribution as well as the
whole electro-weak sector, and that the parameter in Eq. (20.34)
$s_0$ is an a priori unknown scale. However, if we were to assume
a reasonable hadronic scale, $s_0 \simeq 1 \text{GeV}^2$, we would find that
Eq. (20.34) implies an unrealistically high upper bound in com-
parison with the cross sections observed at present collider en-
ergies. Nevertheless there is a common trend in the literature
(see for instance, reviews [225,226] and references therein) to fit
phenomenologically the total cross section with $\ln^2$s, keeping in
mind the saturation of the FLM bound. Such an asymptotic be-
haviour is assumed also by the COMPETE collaboration [22],
which achieved a comprehensive description of all soft pre-LHC
data measured at $\sqrt{s} \ge 4$ GeV as well as total $pp$ cross sections
from the LHC available in the first half of 2015 (see Section 51
in [24]).

It is interesting that the Froissart-type $\ln^2 s$ asymptotics of the $pp$ total cross section are also supported by numerical results in lattice QCD [227]. Such a behaviour is also observed in the approach [228] based on Colour Glass Condensate saturation.

Finally, it is worth mentioning that the possibility that asymp-
totically the Pomeron intercept becomes smaller than 1, $\alpha_P(0) <
1$, and at very high energies the total cross section starts to de-
crease with energy, though highly unlikely, is not yet completely
rejected. For instance, such a behaviour is expected in a theory
with only the triple-Pomeron coupling, $g_{3P}$, and which neglects the
more complicated multi-Pomeron vertices $g_m^n$, such as the $2 \to 2$
Pomeron coupling [229,230].

It was also argued that in the case of an increasing (with energy)
cross section the only regime consistent asymptotically with both
the s- and the t-channel unitarities is that of a black disc whose
radius increases as $R = c \cdot \ln s$ [231] (i.e. $R \propto (\ln s)^\gamma$, with $\gamma = 1$
exactly).

20.8 Acknowledgements

It is a pleasure to thank Michael Albrow, Robert Cahn, Robert
Ciesielski, Per Grafstrom, Frank Krauss, Paul Newman, Sergey

Ostapchenko and Edward Sarkisyan-Grinbaum for discussions,
suggestions and comments on this review. Special thanks to
Graeme Watt who read the article through and helped to im-
prove the presentation. MT is supported by MEYS of the Czech
Republic within project LTT17018. VAK thanks the Institute of
Physics of the Czech Academy of Sciences in Prague for hospitality.

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Astrophysics and Cosmology

<table><tr><td>21. Experimental tests of gravitational theory (rev.)</td><td>431</td></tr><tr><td>22. Big-Bang cosmology (rev.)</td><td>444</td></tr><tr><td>23. Inflation (rev.)</td><td>457</td></tr><tr><td>24. Big-Bang nucleosynthesis (rev.)</td><td>474</td></tr><tr><td>25. Cosmological parameters (rev.)</td><td>482</td></tr><tr><td>26. Neutrinos in cosmology (rev.)</td><td>491</td></tr><tr><td>27. Dark matter (rev.)</td><td>499</td></tr><tr><td>28. Dark energy (rev.)</td><td>515</td></tr><tr><td>29. Cosmic microwave background (rev.)</td><td>524</td></tr><tr><td>30. Cosmic rays (rev.)</td><td>535</td></tr></table>
<figure><img src="image_127.png" /></figure>
# 21. Experimental Tests of Gravitational Theory

Revised August 2023 by T. Damour (IHES, Bures-sur-Yvette).

## 21.1 General Relativity

Einstein's theory of General Relativity (GR), the current "standard" theory of gravitation, describes gravity as a universal deformation of the Minkowski metric:

$$g_{\mu\nu}(x^{\lambda}) = \eta_{\mu\nu} + h_{\mu\nu}(x^{\lambda}), \quad \text{where} \quad \eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1). \qquad (21.1)$$

GR is classically defined by two postulates, embodied in the total action defining the theory:

$$S_{\text{tot}}[g_{\mu\nu}, \psi, A_{\mu}, H] = c^{-1} \int d^4 x (\mathcal{L}_{\text{Ein}} + \mathcal{L}_{\text{SM}}). \qquad (21.2)$$

The first postulate that states that the Lagrangian density describing the propagation and self-interaction of the gravitational field is

$$\mathcal{L}_{\text{Ein}}[g_{\alpha\beta}] = \frac{c^4}{16\pi G} \sqrt{g} g^{\mu\nu} R_{\mu\nu}(g_{\alpha\beta}), \qquad (21.3)$$

where G denotes Newton's constant, $g = -\det(g_{\mu\nu})$, $g^{\mu\nu}$ is the matrix inverse of $g_{\mu\nu}$, and where the Ricci tensor $R_{\mu\nu} \equiv R^{\alpha}_{\mu\alpha\nu}$ is the only independent trace of the curvature tensor

$$R^{\alpha}_{\mu\beta\nu} = \partial_{\beta}\Gamma^{\alpha}_{\mu\nu} - \partial_{\nu}\Gamma^{\alpha}_{\mu\beta} + \Gamma^{\alpha}_{\sigma\beta}\Gamma^{\sigma}_{\mu\nu} - \Gamma^{\alpha}_{\sigma\nu}\Gamma^{\sigma}_{\mu\beta}, \qquad (21.4)$$

$$\Gamma^{\lambda}_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} (\partial_{\mu} g_{\nu\sigma} + \partial_{\nu} g_{\mu\sigma} - \partial_{\sigma} g_{\mu\nu}). \qquad (21.5)$$

The second postulate states that $g_{\mu\nu}$ (and its associated connection) couples universally, and minimally, to all the bosonic (respectively fermionic) fields of the Standard Model by replacing everywhere the Minkowski metric $\eta_{\mu\nu}$ (respectively the flat Minkowski connection). Schematically (suppressing matrix indices and labels for the various gauge fields and fermions and for the Higgs doublet),

$$\mathcal{L}_{\text{SM}}[\psi, A_{\mu}, H, g_{\mu\nu}] = -\frac{1}{4} \sum \sqrt{g} g^{\mu\alpha} g^{\nu\beta} F^{a}_{\mu\nu} F^{a}_{\alpha\beta} - \sum \sqrt{g} \bar{\psi} \gamma^{\mu} (D_{\mu} + \frac{1}{4} \omega_{ij\mu} \gamma^{ij}) \psi - \frac{1}{2} \sqrt{g} g^{\mu\nu} \overline{D_{\mu}} \overline{H} D_{\nu} H - \sqrt{g} V(H) - \sum \lambda \sqrt{g} \bar{\psi} H \psi. \qquad (21.6)$$

Here $F^{a}_{\mu\nu} = \partial_{\mu}A^{a}_{\nu} - \partial_{\nu}A^{a}_{\mu} + g_{A}f^{a}_{bc}A^{b}_{\mu}A^{c}_{\nu}$ and the (representation-dependent) gauge-field covariant derivative $D_{\mu} = \partial_{\mu} + g_{A}A^{a}_{\mu}T^{rep}_{a}$ are defined as in Special Relativity, while the derivative of spin-$\frac{1}{2}$ fermions also includes a coupling to the gravitational “spin-connection” $\omega_{ij\mu} = -\omega_{ji\mu}$, via its contraction with $\gamma^{ij} = \frac{1}{2}(\gamma^{i}\gamma^{j} - \gamma^{j}\gamma^{i})$, where $i, j = 0, 1, 2, 3$ and $\gamma^{i} = e^{i}_{\mu}\gamma^{\mu}$ are usual (numerical) Dirac matrices satisfying $\gamma^{i}\gamma^{j} + \gamma^{j}\gamma^{i} = 2\eta^{ij}$. The connection components $\omega_{ij\mu}$ are defined in terms of the local orthonormal frame (vierbein) $e^{i}_{\mu}$ (such that $g_{\mu\nu} = \eta_{ij}e^{i}_{\mu}e^{j}_{\nu}$) used to describe the components of the various fermions $\psi$, and of its inverse $e^{i}_{\mu}$ (such that $e^{i}_{\mu}e^{j}_{\nu} = \delta^{j}_{i}$), by $\omega_{ij\mu} = \frac{1}{2}(C_{i[jk]} + C_{j[ki]} - C_{k[ij]}) e^{k}_{\mu}$ where $C_{i[jk]} = \eta_{is}C^{s}_{[jk]}$, with $C^{i}_{[jk]} \equiv (\partial_{\mu}e^{i}_{\nu} - \partial_{\nu}e^{i}_{\mu}) e^{j}_{\mu}e^{k}_{\nu}$. From the total action follow Einstein's field equations,

$$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}. \qquad (21.7)$$

Here $R = g^{\mu\nu}R_{\mu\nu}$ is the scalar curvature, and $T_{\mu\nu} \equiv g_{\mu\alpha}g_{\nu\beta}T^{\alpha\beta}$ where $T^{\mu\nu} = (2/\sqrt{g})\delta\mathcal{L}_{\text{SM}}/\delta g_{\mu\nu}$ is the (symmetric) energy-momentum tensor of the Standard Model matter. The theory is invariant under arbitrary coordinate transformations: $x'^{\mu} = f^{\mu}(x^{\nu})$ (as well as under arbitrary local SO(3,1) rotations of the vierbein, $e^{i}_{\mu} = \Lambda^{i}_{j}(x)e^{j}_{\mu}$). To solve the field equations Eq. (21.7), one needs to fix the coordinate gauge freedom, e.g., the “harmonic gauge” (which is the analogue of the Lorenz gauge, $\partial_{\mu}A^{\mu} = 0$, in electromagnetism) corresponds to imposing the condition $\partial_{\nu}(\sqrt{g}g^{\mu\nu}) = 0$.

In this Review, we only consider the classical limit of gravitation (i.e. classical matter and classical gravity). Quantum

gravitational effects are expected (when considered at low energy) to correct the classical action Eq. (21.2) by additional terms involving quadratic and higher powers of the curvature tensor. This suggests that the validity of classical gravity extends (at most) down to length scales of order the Planck length $L_P = \sqrt{\hbar G/c^3} \simeq 1.62 \times 10^{-33}$ cm, i.e., up to energy scales of order the Planck energy $E_P = \sqrt{\hbar c^5/G} \simeq 1.22 \times 10^{19}$ GeV. Considering quantum matter in a classical gravitational background also poses interesting challenges, notably the possibility that the zero-point fluctuations of the matter fields generate a nonvanishing vacuum energy density $\rho_{\text{vac}}$, corresponding to a term $-\sqrt{g} \rho_{\text{vac}}$ in $\mathcal{L}_{\text{SM}}$ [1]. This is equivalent to adding a “cosmological constant” term $+\Lambda g_{\mu\nu}$ on the left-hand side of Einstein’s equations, Eq. (21.7), with $\Lambda = 8\pi G \rho_{\text{vac}}/c^4$. Recent cosmological observations (see the following Reviews) suggest a positive value of $\Lambda$ corresponding to $\rho_{\text{vac}} \approx (2.3 \times 10^{-3} \text{eV})^4$. Such a small value has a negligible effect on the non-cosmological tests discussed below.

## 21.2 Key features and predictions of GR

The definition of GR recalled above makes predictions both about the coupling of gravity to matter, and about the structure of the gravitational field beyond its previously known Newtonian aspects.

### 21.2.1 Equivalence Principle

First, the universal nature of the coupling between $g_{\mu\nu}$ and the Standard Model matter postulated in Eq. (21.6) entails many observable consequences that go under the generic name of “Equivalence Principle”.

A first aspect of the Equivalence Principle is that the outcome of a local non-gravitational experiment, referred to local standards, should not depend on where, when, and in which locally inertial frame, the experiment is performed. This means, for instance, that local experiments should neither feel the cosmological evolution of the Universe (constancy of the “constants”), nor exhibit preferred directions in spacetime (isotropy of space, local Lorentz invariance).

A second aspect of the Equivalence Principle is that the kinetic terms, $g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ or $\bar{\psi}\gamma^{i}e^{i}_{\mu}\partial_{\mu}\psi$, of all the fields of Nature (including the gravitational field itself) are universally coupled to the same curved spacetime metric $g_{\mu\nu}(x) = \eta_{ij}e^{i}_{\mu}e^{j}_{\nu}$. This implies in particular that all massless fields should propagate with the same speed.

A third aspect of the Equivalence Principle is that two (electrically neutral) test bodies dropped at the same location and with the same velocity in an external gravitational field should fall in the same way, independently of their masses and compositions (“universality of free fall” or “Weak Equivalence Principle”). In addition, the study (using the nonlinear structure of GR) of the motion, in an external gravitational field, of bodies having a non-negligible, or even strong, self-gravity (such as planets, neutron stars, or black holes) has shown that the latter property of free-fall universality holds equally well for self-gravitating bodies (“Strong Equivalence Principle”).

A last aspect of the Equivalence Principle concerns various universality features of the gravitational redshift of clock rates. GR predicts that, when intercomparing them by means of electromagnetic signals, two (non gravity-based) clocks located along two different spacetime worldlines should exhibit a universal difference in clock rate that depends on their worldlines, but that is independent of their nature and constitution. For instance, two clocks located at two different positions in a static external Newtonian potential $U(x) = \sum Gm/r$ should exhibit, when intercompared by electromagnetic signals, the difference in clock rate, $\tau_1/\tau_2 = \nu_2/\nu_1 = 1 + [U(x_1) - U(x_2)]/c^2 + O(1/c^4)$, (“universal gravitational redshift of clock rates”). Similarly, the comparison of atomic-transition frequencies when observing on Earth a transition that took place on a far-away galaxy should involve (at lowest order in cosmological perturbations) the universal cosmological redshift factor $1 + z = a(t_{\text{reception}})/a(t_{\text{emission}})$ between the Friedmann scale factors $a(t)$ (see below).
### 21.2.2 Quasi-stationary, weak-field (post-Newtonian) gravity

When applied to quasi-stationary, weak-field gravitational fields, Einstein equations, Eq. (21.7), entail a spacetime structure which predicts deviations from Newtonian gravity of the first post-Newtonian (1PN) order, i.e., fractionally smaller than Newtonian effects by a factor $O(v^2/c^2) \sim O(GM/(c^2r))$. The 1PN-accurate solution of Eq. (21.7) reads (in harmonic gauge)

$$ 
\begin{aligned}
g_{00} &= -1 + \frac{2}{c^2}V - \frac{2}{c^4}V^2 + O\left(\frac{1}{c^6}\right), \\
g_{0i} &= -\frac{4}{c^3}V_i + O\left(\frac{1}{c^5}\right), \\
g_{ij} &= \delta_{ij}\left[1 + \frac{2}{c^2}V\right] + O\left(\frac{1}{c^4}\right),
\end{aligned} 
\quad (21.8) 
$$

where $x^0 = ct$, $i, j = 1, 2, 3$, and where the scalar, $V$, and vector, $V_i$, (retarded) potentials are defined in terms of the sources $\sigma = \frac{T^{00}+T^{ii}}{c^2}$, $\sigma_i = \frac{T^{0i}}{c}$ by

$$ V = \Box_{\text{ret}}^{-1}[-4\pi G\sigma] ; \quad V_i = \Box_{\text{ret}}^{-1}[-4\pi G\sigma_i] . \quad (21.9) $$

In GR the gravitational interaction of $N$ moving point masses (labeled by $A = 1, \dots, N$) is described by a reduced (classical) action that admits a diagrammatic expansion:

$$ S_{\text{reduced}} = S^{\text{free}} + S^{\text{tree-level}} + S^{\text{one-loop}} + \dots \quad (21.10) $$

where the free (special-relativistic) action reads

$$ 
\begin{aligned}
S^{\text{free}} &= -\sum_A \int m_A c \sqrt{-\eta_{\mu\nu} dx_A^\mu dx_A^\nu} \\
&= -\sum_A \int dt m_A c^2 \sqrt{1 - v_A^2/c^2},
\end{aligned} 
\quad (21.11) 
$$

while the tree-level (one-graviton-exchange) interaction term reads

$$ S^{\text{tree-level}} = -\frac{8\pi G}{c^4} \int d^4 x T^{\mu\nu} \Box^{-1} \left(T_{\mu\nu} - \frac{1}{2} T \eta_{\mu\nu}\right) = \int dt L^{(2)}. \quad (21.12) $$

Corresponding to the 1PN-accurate metric of Eq. (21.8), the 1PN-accurate expansion of the latter tree-level, two-body interaction Lagrangian $L^{(2)}$ reads (with $r_{AB} \equiv |\mathbf{x}_A - \mathbf{x}_B|$, $n_{AB} \equiv (\mathbf{x}_A - \mathbf{x}_B)/r_{AB}$)

$$ L^{(2)} = \frac{1}{2} \sum_{A \neq B} \frac{G m_A m_B}{r_{AB}} \left[ 1 + \frac{3}{2c^2} (v_A^2 + v_B^2) - \frac{7}{2c^2} (v_A \cdot v_B) \right. \\ 
\left. - \frac{1}{2c^2} (n_{AB} \cdot v_A) (n_{AB} \cdot v_B) + O\left(\frac{1}{c^4}\right) \right] \quad (21.13) $$

The two-body interactions, Eq. (21.13), exhibit $v^2/c^2$ corrections to Newton's $1/r$ potential induced by spin-2 exchange ("gravito-magnetism"). Consistency at the 1PN level, $v^2/c^2 \sim Gm/rc^2$, requires that one also considers the three-body interactions contained in the one-loop contribution $S^{\text{one-loop}}$, corresponding to terms induced by some of the three-graviton vertices and other non-linearities (terms $O(h^2)$ and $O(hT)$ in Eq. (21.15) below), i.e., to the $O(V^2)$ term in Eq. (21.8):

$$ L^{(3)} = -\frac{1}{2} \sum_{B \neq A \neq C} \frac{G^2 m_A m_B m_C}{r_{AB} r_{AC} c^2} + O\left(\frac{1}{c^4}\right). \quad (21.14) $$

### 21.2.3 Gravitational Waves in GR

The linearized approximation to Einstein's field equations, Eq. (21.7), in harmonic gauge $\partial^\nu(h_{\mu\nu} - \frac{1}{2}h\eta_{\mu\nu}) = 0$ (with $h \equiv \eta^{\mu\nu}h_{\mu\nu}$), reads

$$ \Box h_{\mu\nu} = -\frac{16\pi G}{c^4} \left(T_{\mu\nu} - \frac{1}{2} T \eta_{\mu\nu}\right) + O(h^2) + O(hT). \quad (21.15) $$

Outside of any source (i.e., when $T_{\mu\nu} = 0$), this yields $\Box h_{\mu\nu} = 0$, with $\partial^\nu(h_{\mu\nu} - \frac{1}{2}h\eta_{\mu\nu}) = 0$. The generic linearized solution (modulo the diffeomorphism freedom) of the latter vacuum Einstein equations can be written as (with $k^2 = k \cdot k = \eta_{\mu\nu}k^\mu k^\nu$, $k \cdot x = k_\mu x^\mu$)

$$ h_{\mu\nu}(x) = \int d^4 k \delta(k^2) \epsilon_{\mu\nu}(k) e^{ik \cdot x}, \quad (21.16) $$

where the polarization tensor $\epsilon_{\mu\nu}(k)$ must be transverse ($\epsilon_{\mu\nu}k^\nu = 0$) and traceless ($\eta^{\mu\nu}\epsilon_{\mu\nu} = 0$). In addition, $\epsilon_{\mu\nu}(k)$ can be freely submitted to the gauge freedom $\epsilon'_\mu = \epsilon_{\mu\nu} + \xi_\mu k_\nu + \xi_\nu k_\mu$. This implies that gravitational waves (GW) propagate with the speed of light, and (like electromagnetic waves) have only two independent polarizations. In a frame where, say, $k^\mu = (ck, 0, 0, k)$, the two independent linear polarization tensors can be taken to have components only in the transverse 1-2 plane, of the following form: $\epsilon_{11}^+ = -\epsilon_{22}^+ = \epsilon^+$, with $\epsilon_{12}^+ = \epsilon_{21}^+ = 0$; or $\epsilon_\times^\mu = +\epsilon_{\times 21} = \epsilon^\times$, with $\epsilon_{11}^\times = \epsilon_{22}^\times = 0$. Under a little-group rotation of angle $\theta$ in the 1-2 plane, the two circular polarization amplitudes $\epsilon^\pm(\pm) = \epsilon^+ \mp i\epsilon^\times$ vary as $\epsilon^\prime(\pm) = e^{\pm 2i\theta}\epsilon^\pm(\pm)$, thereby characterizing the helicity-2 nature of GWs.

When solving the inhomogeneous equation Eq. (21.15), taking into account the nonlinear contributions $O(h^2)+O(hT)$, one finds that, to lowest order, the GW amplitude emitted at large distances by a matter distribution is given by the following "quadrupole formula"

$$ h_{ij}^{\text{TT}}(T, \mathbf{X}) \approx \frac{2G}{c^4} P_{ijab}^{\text{TT}}(\mathbf{N}) \frac{\ddot{Q}_{ab}(T - R/c)}{R}, \quad (21.17) $$

where $Q_{ij}(t) = \int d^3x \sigma(t, \mathbf{x}) (x^i x^j - \frac{1}{3}\delta_{ij}\mathbf{x}^2)$ ($a, b, i, j = 1, 2, 3$) is the quadrupole moment of the source, $R = |\mathbf{X}|$ the distance to the source, $\mathbf{N} = \mathbf{X}/R$ the unit direction from the source to the observer, and $P_{ijab}^{\text{TT}}(\mathbf{N}) = (\delta_{ia} - N_i N_a)(\delta_{jb} - N_j N_b) - \frac{1}{2}(\delta_{ij} - N_i N_j)(\delta_{ab} - N_a N_b)$ the transverse-traceless projector onto the 2-plane orthogonal to $\mathbf{N}$.

### 21.2.4 Strong gravitational fields: neutron stars and black holes

The nonlinear structure of Einstein's equations implies many predictions for strong gravitational fields that distinguish GR from Newtonian gravity. For instance, in Newtonian gravity, there is no upper limit to the dimensionless gravitational potential $U/c^2$, with $U$ satisfying Poisson's equation $\Delta U = -4\pi G\rho$, where $\rho$ denotes the Newtonian mass density. By contrast, in GR, the dimensionless surface gravitational potential $GM/(c^2R)$ of a spherically symmetric (perfect fluid) body cannot exceed $\frac{4}{9}$ [2].

Given an equation of state $p = f(\rho)$ modeling the interior of a (cold) spherically symmetric body (say a non-rotating neutron star), Einstein equations, Eq. (21.7), with $T^{\mu\nu} = (\rho + p)u^\mu u^\nu + pg^{\mu\nu}$, and

$$ g_{\mu\nu} dx^\mu dx^\nu = -e^{2\Phi(r)} c^2 dt^2 + \frac{dr^2}{1 - \frac{2GM(r)}{c^2 r}} + r^2 (d\theta^2 + \sin^2 \theta d\phi^2), \quad (21.18) $$

yield the following Tolman-Oppenheimer-Volkoff radial equations:

$$ p'(r) = - \frac{G(\rho + p/c^2)(M(r) + 4\pi r^3 p/c^2)}{r^2(1 - 2GM(r)/(c^2r))}; \quad (21.19) $$

$$ M'(r) = 4\pi r^2 \rho; \quad (21.20) $$

$$ \Phi'(r) = \frac{G(M(r) + 4\pi r^3 p/c^2)}{r^2(1 - 2GM(r)/(c^2r))}. \quad (21.21) $$

In the exterior of the star ($r \ge R$), the metric takes the Schwarzschild form

$$ g_{\mu\nu} dx^\mu dx^\nu = - \left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \frac{dr^2}{1 - \frac{2GM}{c^2 r}} + r^2 \left(d\theta^2 + \sin^2 \theta d\phi^2\right), \quad (21.22) $$

where $M \equiv M(R)$ is the total gravitational mass of the star. GR predicts, for any given $p = f(\rho)$, several (in principle) observable
features of neutron stars, such as: (i) the maximum mass of a neutron star; (ii) the relation between the radius R and the total mass M; (iii) the dimensionless surface gravitational potential $GM/(c^2R)$ (linked to the surface redshift $\sqrt{-g_{00}} = \sqrt{1 - \frac{2GM}{c^2R}}$ measured by an observer at infinity); (iv) the moment of inertia; and (v) the Love number (tidal polarizability). The current uncertainty on the equation of state of a neutron star yields the GR-predicted approximate range for the maximum mass of non-rotating neutron stars $1.5 M_{\odot} \lesssim M_{\text{max}} \lesssim 2.5 M_{\odot}$, and the absolute upper bound $M_{\text{max}} < 3 M_{\odot}$ [3]. The surface gravitational potential of a typical neutron star is $GM/c^2R_{\text{NS}} \simeq 0.17$, which is a factor $\sim 10^8$ higher than the surface potential of the Earth, and a mere factor 3 below the black hole limit $GM/c^2R_{\text{BH}} = \frac{1}{2}$ to be discussed next.

The existence of a maximum mass for a neutron star led Oppenheimer and Snyder [4] to predict that the end point of stellar evolution for sufficiently heavy stars, after exhaustion of all thermonuclear sources of energy, will be what are now called “black holes.” The latter are solutions of Einstein’s equations whose past structure involves a gravitationally collapsing star, but whose presently observable structure is essentially described (for non-rotating black holes) by the vacuum Schwarzschild solution Eq. (21.22). It took many years for theoretical (and mathematical) physicists to understand that the apparent singularity of the Schwarzschild solution at $r = \frac{2GM}{c^2}$ was a coordinate singularity and that the Schwarzschild spacetime was regular at the “black hole horizon”, $R_{\text{BH}} \equiv \frac{2GM}{c^2}$. The rotating analog of the Schwarzschild spacetime is the Kerr black hole [5].

Black holes are outstanding consequences of GR which enjoy many remarkable properties, notably: (i) presence of a one-way surface (the horizon) for all waves and particles; (ii) absence of “hair” (i.e., barring a possible electric charge, their structure is fully described by only two parameters, total mass, M, and total angular momentum, $J \le GM^2/c$); (iii) existence of a spectrum of damped quasi-normal vibrational modes; and (iv) a behavior under external perturbations similar to ordinary physical objects satisfying the laws of (dissipative) thermodynamics. Moreover, though no classical waves or particles can get out of the horizon, black holes are predicted to slowly evaporate via quantum particle creation.

### 21.2.5 Cosmology

To complete our short tour of the main predictions of GR, let us mention that GR offers the current standard framework for describing the large-scale structure of the Cosmos, from the nearly homogeneous Big Bang (and its plausible inflationary beginning) to the current inhomogeneous Universe undergoing an accelerated expansion. The spacetime structure on large (temporal and spatial) scales is well described by a solution of Einstein’s equations of the form

$$ds^2 = -(1 + 2\Phi(t, \mathbf{x})) c^2 dt^2 + 2W_i(t, \mathbf{x}) dtdx^i + a^2(t) ((1 - 2\Psi(t, \mathbf{x})) \delta_{ij} + h_{ij}(t, \mathbf{x})) dx^i dx^j, \quad (21.23)$$

where, after a suitable gauge-fixing [6], $W_i(t, \mathbf{x})$ is transverse, while $h_{ij}(t, \mathbf{x})$ is transverse and traceless. The source $T^{\mu\nu}$ must involve a certain number of postulated ingredients: an inflaton field; the matter of the Standard Model; a dark matter component; and a cosmological constant contribution $T_{\Lambda}^{\mu\nu} = -\rho_{\text{vac}}g^{\mu\nu}$, with $\rho_{\text{vac}} \equiv c^4\Lambda/(8\pi G)$. The scale factor $a(t)$ of the Friedmann background metric $ds_0^2 = -c^2dt^2 + a^2(t)\delta_{ij}dx^i dx^j$ satisfies the GR-predicted Friedmann equations (with vanishing spatial curvature $k=0$),

$$H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho_{\text{tot}}, \quad (21.24)$$

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3} \left( \rho_{\text{tot}} + \frac{3}{c^2} p_{\text{tot}} \right), \quad (21.25)$$

while the scalar $(\Phi(t, \mathbf{x}), \Psi(t, \mathbf{x}))$, vector $(W_i(t, \mathbf{x}))$, and tensor $(h_{ij}(t, \mathbf{x}))$ inhomogeneous perturbations satisfy some GR-predicted propagation equations (coupled to matter perturbations); see [6] and the following Reviews. When the cosmic fluid is

well approximated by a perfect fluid, Einstein’s equations predict the following link between the scalar perturbations

$$\Phi(t, \mathbf{x}) = \Psi(t, \mathbf{x}). \quad (21.26)$$

## 21.3 A roadmap of parametrizations of deviations from GR, and of modified gravity

As will be discussed below, all currently performed gravitational experiments are compatible with GR. However, similarly to what is done in discussions of precision electroweak experiments, it is useful to quantify the significance of precision gravitational experiments by parameterizing possible deviations from GR. One can distinguish two main approaches to considering, and parameterizing, deviations from GR: (i) theory-agnostic phenomenological approaches; or, (ii) the study of the predictions of specific classes of alternative theories of gravity. Both types have led to useful ways of discussing tests of gravity. Both types also have their limitations. Considering them together leads to cross-fertilization.

### 21.3.1 Theory-agnostic phenomenological approaches to parameterizing deviations from GR

The theory-agnostic phenomenological approach is the oldest, and, arguably, the most robust one. It essentially consists in starting from specific observable predictions within the considered standard theory, and of deforming them by introducing some free parameters measuring either deviations from effects already present within the standard theory, or new effects absent from the standard theory. A classic example is the periastron advance of Mercury (and the other planets). When working within Newtonian gravity as a standard theory of gravity, the rate of periastron advance of Mercury, $\dot{\omega}$, is (when neglecting the quadrupole moment of the Sun) a calculable function of the masses and semi-major axes of the other planets of the solar system, say $\dot{\omega}^{\text{Newton}}(m_i, a_i)$. However, $\dot{\omega}$ is also a directly observable quantity, so that one can parameterize the periastron advance of Mercury by writing

$$\dot{\omega}^{\text{obs}} = \dot{\omega}^{\text{Newton}}(m_i, a_i) + \Delta\dot{\omega}. \quad (21.27)$$

Using other observable data to determine some “observed” values of the $m_i$’s and $a_i$’s, one can then measure the anomalous periastron precession $\Delta\dot{\omega}$ and see whether it is compatible with zero, or not. As is well-known, Leverrier used such a methodology and, in 1859, measured an anomalous periastron precession of about $\Delta\dot{\omega} \simeq 38$ arcsec/century (later re-estimated at 43 arcsec/century), which was explained in 1915 as a GR prediction. Let us discuss further examples of the use of such theory-agnostic approaches for discussing deviations from GR.

### 21.3.2 Parameterized post-Newtonian (PPN) formalism.

When considering the weak-field slow-motion limit appropriate to describing gravitational experiments in the solar system, it has been traditional to parameterize possible (long-range) deviations from the GR-predicted 1PN metric by introducing extra dimensionless coefficients in the various terms of the metric of Eq. (21.8). The minimal version of the parameterized post-Newtonian (PPN) formalism (essentially due to Eddington) involves only two parameters $\beta$ and $\gamma$, namely

$$g_{00} = -1 + \frac{2}{c^2} V - \frac{2\beta}{c^4} V^2 + O\left(\frac{1}{c^6}\right), \quad (21.28)$$

$$g_{0i} = -\frac{2(\gamma + 1)}{c^3} V_i + O\left(\frac{1}{c^5}\right), \quad (21.29)$$

$$g_{ij} = \delta_{ij} \left[ 1 + \frac{2\gamma}{c^2} V \right] + O\left(\frac{1}{c^4}\right), \quad (21.30)$$

with $V$ and $V_i$ defined by Eq. (21.9), with the same vectorial source $\sigma_i = \frac{T^{0i}}{c}$, but a modified scalar source

$$\sigma^{\text{PPN}} = \frac{1}{c^2} \left( \left[ 1 + (3\gamma - 2\beta - 1) \frac{V}{c^2} \right] T^{00} + \gamma T^{ii} \right). \quad (21.31)$$

In GR, $\beta^{\text{GR}} = 1$ and $\gamma^{\text{GR}} = 1$, so that deviations from GR are parameterized by $\bar{\beta} \equiv \beta - 1$ and $\bar{\gamma} \equiv \gamma - 1$. Richer versions of the
PPN formalism (involving up to ten parameters) were developed in interaction with the study of classes of alternative theories of gravity [7, 8]. This led to parameterizing new types of contributions to the 1PN metric that are absent in the GR framework.

When deriving the 1PN-accurate dynamics of N point masses predicted by the PPN-modified metric, Eq. (21.28), one finds that the free Lagrangian is not modified (because we are considering here a Lorentz-invariant subclass of PPN metrics), while there are modifications of both the two-body Lagrangian, $L^{(2)}$, Eq. (21.13), and the three-body one, $L^{(3)}$, Eq. (21.14). More precisely, denoting $\eta \equiv 4\bar{\beta} - \bar{\gamma}$, the Newtonian interaction energy term in Eq. (21.13) is modified into $G_{AB}m_A m_B/r_{AB}$, with a body-dependent gravitational “constant”

$$G_{AB} = G[1 + \eta(E_A^{\text{grav}}/m_A c^2 + E_B^{\text{grav}}/m_B c^2) + O(1/c^4)], \quad (21.32)$$

where $E_A^{\text{grav}}$ denotes the gravitational binding energy of body A. In addition, there is the additional contribution $+\bar{\gamma}(v_A - v_B)^2/c^2$ in the brackets on the right-hand side of $L^{(2)}$, Eq. (21.13). As for the three-body interaction term $L^{(3)}$, Eq. (21.14), it is modified by the overall factor $1 + 2\bar{\beta}$.

These results show how the introduction of the two minimal PPN deviation parameters $\bar{\beta} \equiv \beta - 1$ and $\bar{\gamma} \equiv \gamma - 1$ suffices to introduce many different observable effects. Some of them (the ones linked with $\bar{\gamma}$) concern deviations at the linearized (one-graviton-exchange) level (and affect, for instance, light deflection and time-delay effects), while the deviation parameter $\bar{\beta}$ parameterizes effects linked to the cubic vertex of Einstein's gravity (and affects, for instance, periastron precession). Of particular interest is the fact that Eq. (21.32) shows that the combination $\eta \equiv 4\bar{\beta} - \bar{\gamma}$ parameterizes a violation of the Strong Equivalence Principle, because the gravitational interaction between self-gravitating bodies is seen to be influenced by the gravitational binding energy of each body [9]. As stated above, this effect is absent in GR (where $\eta^{\text{GR}} = 0$). This is an example where the fact of contrasting GR with some deviations from it gives physical significance to a null effect in GR (namely the universality of free fall of self-gravitating bodies).

Finally, one can extend the PPN formalism by allowing for a slow, phenomenological time variation of Newton's constant:

$$G(t) = G_0 \left[ 1 + \frac{\dot{G}_0}{G_0} (t - t_0) \right] . \quad (21.33)$$

Here, one assumes that there exist units in which the masses, $m_i$, of elementary particles stay constant, and that $G$ is measured in such units. A possible time variation of $G$ then corresponds to a possible common variation of the dimensionless couplings $Gm_i^2/(\hbar c)$.

### 21.3.3 Parameterized post-Keplerian (PPK) formalism.

The discovery of pulsars (i.e., rotating neutron stars emitting a beam of radio noise) in gravitationally bound orbits [10, 11] has given us our first experimental handle on a regime of relativistic gravity going significantly beyond the uniformly weak-field, and quasi-stationary regime of solar-system gravity. Binary pulsars allow us to probe some radiative effects, and also some strong-gravitational-field effects. In these systems, the finite speed of propagation of the gravitational interaction between the pulsar and its companion generates damping-like terms at order $(v/c)^5$ in the equations of motion [12]. These damping forces are the local counterparts of the gravitational radiation emitted at infinity by the system (“gravitational radiation reaction”). They cause the binary orbit to shrink and its orbital period $P_b$ to decrease. The remarkable stability of pulsar clocks has allowed one to measure the corresponding very small orbital period decay $\dot{P}_b \equiv dP_b/dt \sim -(v/c)^5 \sim -10^{-12}-10^{-14}$ in several binary systems, thereby giving us a direct experimental handle on the propagation properties of the gravitational field. In addition, the large surface gravitational potential of a neutron star allows one to probe the quasi-static strong-gravitational-field regime, as is discussed below.

It is possible to extract phenomenological (theory-independent) tests of gravity from binary pulsar data by using the parameter-

ized post-Keplerian (PPK) formalism [13]. The basis of this formalism is the fact that, after correcting for the Earth's motion around the Sun and for the dispersion due to propagation in the interstellar plasma, the time of arrival of the Nth pulse $t_N$ can be described by a generic, parameterized “timing formula” [13, 14], whose functional form is common to the whole class of tensor-scalar gravitation theories:

$$t_N - t_0 = F[T_N(\nu_p, \dot{\nu}_p, \ddot{\nu}_p); \{p^K\}; \{p^{PK}\}] . \quad (21.34)$$

Here, $T_N$ is the pulsar proper time corresponding to the Nth turn given by $N/2\pi = \nu_p T_N + \frac{1}{2}\dot{\nu}_p T_N^2 + \frac{1}{6}\ddot{\nu}_p T_N^3$ (with $\nu_p \equiv 1/P_b$ the spin frequency of the pulsar, etc.), $\{p^K\} = \{P_b, T_0, e, \omega_0, x\}$ is the set of “Keplerian” parameters (notably, orbital period $P_b$, eccentricity $e$, periastron longitude $\omega_0$ and projected semi-major axis $x = a \sin i/c$), and $\{p^{PK}\} = \{k, \gamma_{\text{timing}}, \dot{P}_b, r, s, \delta_\theta, \dot{\epsilon}, \ddot{\epsilon}\}$ denotes the set of (separately measurable) “post-Keplerian” parameters. Most important among these are: the fractional periastron advance per orbit $k \equiv \dot{\omega} P_b / 2\pi$; a dimensionful time-dilation parameter $\gamma_{\text{timing}}$; the orbital period derivative $\dot{P}_b$; and the “range” and “shape” parameters of the gravitational time delay caused by the companion, $r$ and $s$.

Without assuming any specific theory of gravity, one can phenomenologically analyze the data from any binary pulsar by least-squares fitting the observed sequence of pulse arrival times to the timing formula of Eq. (21.34). This fit yields the “measured” values of the parameters $\{\nu_p, \dot{\nu}_p, \ddot{\nu}_p\}$, $\{p^K\}$, $\{p^{PK}\}$. Now, each specific relativistic theory of gravity predicts that, for instance, $k, \gamma_{\text{timing}}, \dot{P}_b, r$, and $s$ (to quote parameters that have been successfully measured from some binary pulsar data) are some theory-dependent functions of the Keplerian parameters and of the (unknown) masses $m_1, m_2$ of the pulsar and its companion. For instance, in GR, one finds (with $M \equiv m_1 + m_2, n \equiv 2\pi/P_b$),

$$ \begin{aligned} k^{\text{GR}}(m_1, m_2) &= 3(1 - e^2)^{-1} (GMn/c^3)^{2/3}, \\ \gamma_{\text{timing}}^{\text{GR}}(m_1, m_2) &= en^{-1} (GMn/c^3)^{2/3} m_1 (m_1 + 2m_2)/M^2, \\ \dot{P}_b^{\text{GR}}(m_1, m_2) &= -(192\pi/5)(1 - e^2)^{-7/2} \left(1 + \frac{73}{24}e^2 + \frac{37}{96}e^4\right) \\ & \quad \times (GMn/c^3)^{5/3} m_1 m_2/M^2, \\ r^{\text{GR}}(m_1, m_2) &= Gm_2/c^3, \\ s^{\text{GR}}(m_1, m_2) &= nx(GMn/c^3)^{-1/3} M/m_2. \end{aligned} \quad (21.35) $$

In alternative gravity theories each of the functions $k^{\text{theory}}(m_1, m_2)$, $\gamma_{\text{timing}}^{\text{theory}}(m_1, m_2)$, $\dot{P}_b^{\text{theory}}(m_1, m_2)$, etc., is modified by quasi-static strong field effects (associated with the self-gravities of the pulsar and its companion), while the particular function $\dot{P}_b^{\text{theory}}(m_1, m_2)$ is further modified by radiative effects [15–18]. If one measures $N > 2$ PPK parameters from the data of a specific binary pulsar, these $N$ measurements determine, for each given theory, $N$ curves (defined by the $N$ equations $k_i^{\text{theory}}(m_1, m_2) = k_i^{\text{obs}}$) in the two-dimensional mass plane $(m_1, m_2)$. This yields $N - 2$ tests of the specified theory, according to whether the $N$ curves (or strips) have one point in common, as they should.

### 21.3.4 Parameterized-post-Friedmannian (PPF) formalisms.

We have recalled above that, in GR, the two functions, $\Phi(t, \mathbf{x})$, and $\Psi(t, \mathbf{x})$, parameterizing (in the “longitudinal gauge”) the scalar perturbations of the background Friedmann metric are related (in absence of anisotropic stresses) by Eq. (21.26). Several authors [19–28] have defined various types of parameterized-post-Friedmannian (PPF) formalisms involving (generally space and time dependent) phenomenological parameters. The simplest versions of these formalisms involve two phenomenological parameters measuring: (i) the ratio between $\Phi(t, \mathbf{x})$, and $\Psi(t, \mathbf{x})$, say (using a parametrization which parallels the usual PPN parametrization)

$$\Psi(t, \mathbf{x}) = \gamma_{\text{cosmo}}(t, \mathbf{x})\Phi(t, \mathbf{x}); \quad (21.36)$$
and (ii) the effective gravitational constant entering the Poisson equation for $\Phi(t, \mathbf{x})$, say

$$ \Delta\Phi(t, \mathbf{x}) = 4\pi G_{\Phi}(t, \mathbf{x})\delta\rho(t, \mathbf{x}). \quad (21.37) $$

However, the peculiarities of cosmological observables limit the domain of applicability of such phenomenological approaches [26] (notably because the strong dependence of cosmological probes on epochs and scales obliges one to rely on specific parameterizations of the functions $\gamma_{\text{cosmo}}(t, \mathbf{x})$ and $G_{\Phi}(t, \mathbf{x})$, e.g., [25, 28]). Approaches based on specific classes of modified-gravity theories allow for a more complete treatment involving, in principle, all existing cosmological observables: Big Bang nucleosynthesis, cosmic microwave background, large-scale structure, Hubble diagram, weak lensing, etc. Discussing the current cosmological tests using either such PPF formalisms, or comparisons with the predictions of modified-gravity theories, is beyond the scope of this review. See [29] for a comprehensive recent discussion. The bottom line is that all present cosmological data have been found to be compatible with GR (within the Friedmann-Lemaître-based ΛCDM model). Beyond the quantitative limits on various parameterized theoretical models [29], one should remember the striking (strong-field-type) qualitative verification of GR embodied in the fact that relativistic cosmological models give an accurate picture of the Universe over a period during which the spatial metric has been blown up by a gigantic factor, say $(1+z)^2 \sim 10^{19}$ between Big Bang nucleosynthesis and now.

### 21.3.5 Various phenomenological tests of GR from gravitational wave (GW) data

The observations by the US-based Laser Interferometer Gravitational-wave Observatory (LIGO), later joined by the Europe-based Virgo detector, of gravitational-wave (GW) signals [30–34], have opened up a novel testing ground for relativistic gravity. The first three observing runs (O1, O2, O3a and O3b) of the LIGO-Virgo collaboration, recently joined by the Japan-based KAGRA detector (LVK collaboration), have led to the confident detection of 93 GW signals, most of which come from binary black-hole coalescences. These observations are summarized in three GW transient catalogs: GWTC-1 [35], GWTC-2 [36] and GWTC-3 [37].

Several approaches have been used to either test consistency with GR, or to look for special types of possible deviations. Making accurate predictions for GW signals from coalescing black holes within GR took years of both analytical [38, 39] and numerical [40] work. Some works have started (both analytically [15, 41–45] and numerically [46–48]) to derive the corresponding predictions within some modified-gravity theories. Phenomenological approaches are very useful for parameterizing general, conceivable deviations from GR when analyzing the GW signals emitted by coalescing black holes or neutron stars.

A first phenomenological, global consistency test simply consists of measuring the noise-weighted correlation $C$ between each detected strain signal and the corresponding best-fit GR-predicted waveform. $C^{\text{obs}}$ should be equal to 1, modulo statistical (and/or systematic) errors.

Various other phenomenological tests of the structure of the GR-predicted waveforms emitted by coalescing compact binaries have been suggested. One general idea [49–51] (dubbed “parameterized post-Einsteinian formalism” in [52]) is to modify the GR-predicted Fourier-domain value $\psi(f)$ of the phase of black-hole coalescence GW signals $h(f) = A(f)e^{i\psi(f)}$ by introducing GR-deviation parameters, say

$$ \psi(f) = \sum_{i} \left[ p_{i}^{\text{GR,NS}}(m_1, m_2)(1 + \delta\hat{p}_i) + p_{i}^{\text{GR,S}}(m_1, m_2, S_1, S_2) \right] \times u_i(f). \quad (21.38) $$

Here, the $u_i(f)$'s define a basis of functions of the GW frequency $f$, and the superscript NS refers to the nonspinning contribution, while the superscript S refers to the spinning one. Such a GR-modification directly applies to the phenomenological representation [53] of $\psi(f)$, and can be generalized to any waveform model by adding the non-GR phase term $\sum_i p_i^{\text{GR,NS}}(m_1, m_2)\delta\hat{p}_i u_i(f)$

to the corresponding GR-predicted Fourier-domain phase $\psi(f)$ [54]. For instance, the leading-order (LO), quadrupolar term in the GR phase evolution during the early inspiral corresponds to $u_0(f) = f^{-5/3}$ and $p_0^{\text{GR}} = \frac{3(m_1+m_2)^2}{128m_1m_2}(\pi G(m_1+m_2)/c^3)^{-5/3}$, while the next-to-leading-order (NLO) term is a $O(v^2/c^2)$ correction $p_2^{\text{GR}}u_2(f)$ with $u_2(f) = f^{-1}$. In the phenomenological model [53] these terms (as well as the other inspiral contributions) are cut-off beyond the frequency $G(m_1+m_2)/c^3 = 0.018$.

Each dimensionless parameter $\delta\hat{p}_i$ introduces a fractional deviation from the corresponding individual phasing GR effect having the frequency dependence $u_i(f)$, and can, in principle, be extracted by fitting the inspiral part of the observed waveform to the deformed template for simultaneously extracting the values of $m_1, m_2, S_1$, and $S_2$. Together with signal-to-noise ratio (SNR) considerations, and parameter-correlation issues, this limits the applicability of such a test to introducing only one deformation parameter $\delta\hat{p}_i$ at a time. A particularly meaningful test [49] is to leave undeformed the LO and NLO terms $p_0^{\text{GR}}u_0(f) + p_2^{\text{GR}}u_2(f)$ and to vary the third coefficient $p_3$ parameterizing the next, “GW tail”-related $O(v^3/c^3)$ correction, with $u_3(f) = f^{-2/3}$. Another well-motivated test [52] is to introduce a new coefficient $\delta\hat{p}_{-2}$, which is absent in GR, and which parameterizes an $O((v/c)^{-2})$ fractional correction to the LO, quadrupolar term, thereby allowing for a possible dipolar GW flux (indeed, dipolar GW radiation generally exists in theories containing scalar excitations). As $p_2^{\text{GR}}$ vanishes, $\delta\hat{p}_{-2}$ is added as an absolute deviation, scaled by the LO term $p_0^{\text{GR}}$.

The coalescence of two black holes, or of a black hole and a neutron star (or of two heavy-enough neutron stars) leads to the formation of a black hole that is initially formed in a perturbed state. The relaxation of the latter perturbed black hole into its stationary, equilibrium state leads to the emission of characteristic (rapidly decaying) ringing GW modes (a.k.a. quasi-normal modes) [55, 56], whose frequencies and decay times are functions of the mass ($M_f$) and spin ($S_f \equiv GM_f^2 a_f/c$) of the final black hole, say

$$ \omega_a = (c^3/GM_f)[2\pi\hat{f}_a^{\text{QNM}}(a_f) - i\hat{\tau}_a^{\text{QNM}}(a_f)], \quad (21.39) $$

where $a = 1, 2, \dots$ labels the various ringing modes, starting from the least-damped one. In principle, if the SNR is large enough, one can directly test for the presence of one or several of these modes in the post-merger signal, and measure both $\text{Re}(\omega_a)$ and $\text{Im}(\omega_a)$ in a theory-independent way. These phenomenological measurements then lead to null tests of GR, from which one can extract theoretical information about eventual deviations from GR [57, 58].

As recalled above, GR predicts that GWs propagate (in vacuum) at exactly the same speed as light (i.e., they have the same dispersion law $g^{\mu\nu}k_\mu k_\nu = 0$ in curved spacetime). Deviations from such a universal, scale-free dispersion law can be phenomenologically parameterized in several ways. If one phenomenologically assumes that the graviton dispersion law includes a mass term, say $g^{\mu\nu}k_\mu k_\nu + m_g^2/\hbar^2 = 0$, or some more general type of frequency-dependent modification, such changes affect the phasing of the inspiral GW signal and can be directly tested [59]. When one observes both GWs and electromagnetic waves emitted by the same system, one can also directly test whether both types of waves propagate in the same way.

Let us now present some examples of theory-dependent discussions of experimental tests based on considering specific classes of alternative theories. The most conservative deviations from Einstein's pure spin-2 theory are defined by adding new, bosonic, light or massless, macroscopically coupled fields.

### 21.3.6 Gravity tests within classes of tensor-scalar theories of gravity

The possible existence of new gravitational-strength couplings leading to deviations from Einsteinian (and Newtonian) gravity has been suggested by many natural extensions of GR, starting with the classic Kaluza-Klein idea, and continuing up to now with the study of extended supergravity theories, and of (super-
)string theory. In particular, a recurrent suggestion of such theories (which dates back to pioneering work by Jordan, and by Fierz [60]) is the existence of a scalar field $\varphi$ coupled both to the scalar curvature $R$ and to the various $F_{\mu\nu}^{a2}$ gauge-field actions. Such fields (“dilaton” or “moduli”) generically appear in string theory and are massless at the tree-level, but could acquire a self-interaction potential $V(\varphi)$ beyond the tree-level.

The exchange of such a dilaton-like field leads to several types of observational deviations from GR. For experimental limits on the gravitational inverse-square-law (down to the micrometer range) see Refs. [61–65]. If the potential $V(\varphi)$ is zero or negligible for the considered range, the coupling of $\varphi$ to $F_{\mu\nu}^{a2}$ leads to apparent violations of the weak equivalence principle, with rather specific composition-dependence [66]. Next, when neglecting the fractionally small composition-dependent effects, such a field approximately couples to the trace of the energy-momentum tensor $T = g_{\mu\nu}T^{\mu\nu}$. The most general such theory contains (after suitable field redefinitions) two arbitrary functions of the scalar field, namely the self-interaction potential $V(\varphi)$, and a matter-coupling function $a(\varphi)$:

$$ \mathcal{L}_{\text{tot}}[g_{\mu\nu}, \varphi, \psi, A_\mu, H] = \frac{c^4}{16\pi G_*} \sqrt{\bar{g}}(R(g_{\mu\nu}) - 2g^{\mu\nu} \partial_\mu \varphi \partial_\nu \varphi) - \sqrt{\bar{g}}V(\varphi) + \mathcal{L}_{\text{SM}}[\psi, A_\mu, H, \tilde{g}_{\mu\nu}]. \quad (21.40) $$

Here $G_*$ is a “bare” Newton constant, and the Standard Model matter is coupled not to the “Einstein” (pure spin-2) metric $g_{\mu\nu}$, but to the conformally related (“Jordan-Fierz”) metric

$$ \tilde{g}_{\mu\nu} = \exp(2a(\varphi))g_{\mu\nu}. \quad (21.41) $$

The scalar field equation

$$ \Box_g \varphi = \frac{4\pi G}{c^4} \left( -\alpha(\varphi)T + \frac{\partial V(\varphi)}{\partial \varphi} \right), \quad (21.42) $$

features

$$ \alpha(\varphi) \equiv \partial a(\varphi)/\partial \varphi, \quad (21.43) $$

as the basic (field-dependent) coupling between $\varphi$ and matter [15, 67]. The best-known, special case of these theories is the one-parameter ($\omega$) Jordan-Fierz-Brans-Dicke theory [68], with $V(\varphi) = 0$ and $a(\varphi) = \alpha_0\varphi$, leading to a field-independent coupling $\alpha(\varphi) = \alpha_0$ (with $\alpha_0^2 = 1/(2\omega + 3)$). More generally, if we consider the massless theories ($V(\varphi) = 0$) with arbitrary (non-linear) coupling function $a(\varphi)$, they modify Einstein’s predictions in the weak-field slow-motion limit appropriate to describing gravitational experiments in the solar system (1PN approximation) only through the appearance of exactly the same two “post-Einstein” dimensionless parameters $\bar{\gamma} = \gamma - 1$ and $\bar{\beta} = \beta - 1$ that entered the minimal (Eddington) PPN formalism presented above. However, we now have the following theoretical expressions relating the latter phenomenological parameters to the coupling functions entering the tensor-scalar action Eq. (21.40):

$$ \bar{\gamma} = -2 \frac{\alpha_0^2}{1 + \alpha_0^2}; \quad (21.44) $$

$$ \bar{\beta} = +\frac{1}{2} \frac{\beta_0 \alpha_0^2}{(1 + \alpha_0^2)^2}. \quad (21.45) $$

Here $\alpha_0 \equiv \alpha(\varphi_0)$, and $\beta_0 \equiv \partial\alpha(\varphi_0)/\partial\varphi_0$, with $\varphi_0$ denoting the vacuum expectation value (VEV) of $\varphi$ around the solar system. In addition, the observable value $G^{\text{obs}}$ of the gravitational constant is found to be field-dependent and given (at a place where $\varphi = \varphi_0$) by

$$ G^{\text{obs}} = G(\varphi_0) \equiv G_* \exp[2a(\varphi_0)](1 + \alpha_0^2). \quad (21.46) $$

This makes it clear that the parameter $\bar{\gamma}$ is the basic post-Einstein parameter, which measures the admixture of an additional field (here a spin-0 field) to the pure spin-2 GR. One also sees how the parameter $\bar{\beta}$ is linked to non-linear effects (here coupling terms

$\beta_0(\varphi - \varphi_0)^2 T$ in the action), and how the Nordtvedt parameter $\eta \equiv 4\bar{\beta} - \bar{\gamma}$ is related to the field-dependence of $G^{\text{obs}}$ ($\eta = (\alpha_0/(1+\alpha_0^2))\partial\ln G(\varphi_0)/\partial\varphi_0$).

The advantage of a theory-dependent approach, such as Eq. (21.40), over the phenomenological minimal PPN approach of Eq. (21.28), is that it allows one to consistently predict the observational deviations from GR in all possible gravity regimes: the quasi-stationary weak-field regime; the wavelike weak-field regime; the strong-field regime; the cosmological regime, etc. All such observational deviations can be consistently worked out once one chooses specific forms of the coupling function $a(\varphi)$, and of $V(\varphi)$. The simple choice of a two-parameter quadratic coupling function, say $a(\varphi) = \alpha_0(\varphi - \varphi_0) + \frac{1}{2}\beta_0(\varphi - \varphi_0)^2$, has been found useful for describing many possible observable deviations from GR.

The observable consequences for binary pulsar observations of the strong-field and radiative effects linked to the coupling to $\varphi$ have been explicitly worked out in Refs. [15, 44] in the case where $\varphi$ is massless (see Ref. [69] for the case where $\varphi$ is massive). In particular, the strong-field nature of the pulsar tests is demonstrated by the fact that some tensor-scalar theories can be as close as desired to GR in the weak-field regime of the solar-system (i.e., $\bar{\gamma}$ and $\bar{\beta}$ can be as small as desired, or even exactly zero), while developing (via a “spontaneous scalarization” mechanism) differences of order unity with GR in binary pulsar experiments [17, 18].

### 21.3.7 Attractor and screening mechanisms in modified gravity

As will follow from the discussion of experimental data below, the comparison between the predictions of general massless tensor-scalar theories and current data shows that the basic coupling parameter $\alpha_0$ must be tuned to a small value (especially when allowing for composition-dependent effects). This raises the issue of the naturalness of such small coupling parameters. It has been shown in this respect that, in many tensor-scalar theories, there is an *attractor mechanism* by which the cosmological evolution naturally drives the VEV $\varphi_0(t)$ towards a value for which the coupling parameter $\alpha_0 = \alpha(\varphi_0)$ vanishes, thereby making it natural to expect only small deviations from GR (at least for the weak-field regime) at our current cosmological epoch [70, 71].

There are other theoretical mechanisms (generically called “screening mechanisms”) that could explain why a theory of gravity whose theoretical content significantly differs from that of GR could naturally pass all the stringent, GR-compatible experimental limits that will be discussed below. In particular, when considering a self-interacting scalar field ($V(\varphi) \neq 0$), the interplay between the two terms on the right-hand side of Eq. (21.42) tends to drive the local VEV $\varphi_0$ of $\varphi$ to a density-dependent value. In turn, this leads to a corresponding density-dependent effective mass $m_0(\varphi_0) = \sqrt{4\pi G \partial^2 V(\varphi_0)/\partial\varphi_0^2}$ of the $\varphi$ field, and to density-dependent matter couplings [72]. Various choices of the functions $V(\varphi)$ and $a(\varphi)$ can then reduce the $\varphi$-induced deviations from GR in dense environments while still allowing for significant deviations in different (e.g., cosmological) regimes [73–77].

Other screening mechanisms have been invoked, based on an environment dependence mediated by (first or second) derivatives of a scalar degree of freedom. Roughly speaking, such mechanisms involve a (possibly effective) scalar degree of freedom $\varphi$ that satisfies a field equation that is more general than Eq. (21.42) in that the left-hand side, $\Box_g\varphi$, is replaced by a non-linear function of $\varphi$, $\partial\varphi$ and $\partial^2\varphi$. The presence of non-linear derivative self-interactions of $\varphi$ can weaken the effective coupling of $\varphi$ to matter. A simple toy-model showing this weakening would be to replace Eq. (21.42) by an equation of the form

$$ Z(\varphi, \partial\varphi, \partial^2\varphi) \Box_g \varphi = \frac{4\pi G}{c^4} \left( -\alpha(\varphi)T + \frac{\partial V(\varphi)}{\partial\varphi} \right). \quad (21.47) $$

Such an equation is equivalent, at a first level of approximation, to replacing the gravitational constant $G$ entering Eq. (21.42) by $G_{\text{eff}}(\varphi_0, \partial\varphi_0, \partial^2\varphi_0) \equiv G/Z_0$, where $Z_0 \equiv Z(\varphi_0, \partial\varphi_0, \partial^2\varphi_0)$. This has a screening effect if $Z_0 \gg 1$. Indeed, the replacement $G \to G_{\text{eff}}$ diminishes the strength of the interaction potential due to $\varphi$ exchange by a factor of $1/Z_0$. In addition, the range of this
interaction is also affected: $m_0(\varphi_0) = \sqrt{4\pi G \partial^2 V(\varphi_0)/\partial \varphi_0^2} \rightarrow m_{0\text{eff}}(\varphi_0, \partial\varphi_0, \partial^2\varphi_0) = \sqrt{4\pi G_{\text{eff}} \partial^2 V(\varphi_0)/\partial\varphi_0^2} = Z_0^{-1/2} m_0$.

Screening mechanisms based on such non-linear derivative self-interactions are often referred to as being “Vainshtein-like” because a similar mechanism was first invoked in Ref. [78] as a conjectural way to ensure that the extra degrees of freedom associated with a massive (rather than massless) graviton become effectively weakly coupled to matter within a large domain around gravitational sources. Here, one is considering massive deformations of the massless spin-2 metric field of GR by a very small mass, possibly of cosmological scale: $m_g \sim \hbar H_0 \sim 10^{-33}$ eV. The construction of ghost-free potential terms for a spin-2 field has turned out to be a delicate matter [79]. The phenomenology of a very-low-mass graviton is still partly uncontrolled, both because of the unknown extent to which the Vainshtein screening is really active, and because of subtle constraints linked to an eventual UV completion of the theory beyond the unusually low energy scale where it becomes strongly coupled:

$$ \Lambda_{\text{strong coupling}} \sim (M_{\text{Planck}}m_0^2)^{1/3} \sim 10^{-13} \left( \frac{m_0}{\hbar H_0} \right)^{2/3} \text{eV}. \qquad (21.48) $$

The search for modified gravity theories incorporating an extra scalar degree of freedom potentially able to yield a Vainshtein-like screening led to writing down the following general class of tensor-scalar Lagrangian [80,81]:

$$ \begin{aligned} L_{\text{tot}}[g_{\mu\nu}, \varphi, \psi] = & G_2(\varphi, X) - G_3(\varphi, X) \Box_g \varphi + G_4(\varphi, X) R \\ & + G_{4X}(\varphi, X) [(\Box_g \varphi)^2 - \varphi^{\mu\nu} \varphi_{\mu\nu}] \\ & + G_5(\varphi, X) G^{\mu\nu} \varphi_{\mu\nu} - \frac{1}{6} G_{5X}(\varphi, X) [(\Box_g \varphi)^3 \\ & - 3 \Box_g \varphi \varphi^{\mu\nu} + 2 \varphi_{\mu\nu} \varphi^{\mu\lambda} \varphi_{\lambda}^{\nu}] + L_{\text{matter}}[g_{\mu\nu}, \psi]. \end{aligned} \qquad (21.49) $$

Here $g_{\mu\nu}$ denotes the matter-coupled metric, $X \equiv -\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi$, $\varphi_{\mu\nu} \equiv \nabla_{\mu}\nabla_{\nu}\varphi$, $G^{\mu\nu} \equiv R^{\mu\nu} - \frac{1}{2}Rg^{\mu\nu}$, and the various coefficients $G_n(\varphi, X)$ are arbitrary functions of two variables (with $G_n X \equiv \partial G_n / \partial X$). The field equations derived from the Lagrangian of Eq. (21.49) are only of second order in derivatives in spite of the non-linear structure of $L_{\text{tot}}$. This implies that the tensor-scalar theories defined by Eq. (21.49) feature three degrees of freedom, corresponding to a massless spin-2 excitation (GW) and a spin-0 excitation. Contrary to the simpler tensor-scalar theories of Eq. (21.40), it is found that the speed of propagation of GWs implied by Eq. (21.49) is generically different from the speed of light:

$$ \frac{c_{\text{GW}}^2}{c^2} = \frac{G_4 - X(\ddot{\varphi}G_{5X} + G_{5\varphi})}{G_4 - 2XG_{4X} - X(H\ddot{\varphi}G_{5X} - G_{5\varphi})}. \qquad (21.50) $$

More general modified gravity models have been proposed (see, e.g. Refs. [82,83]). Apart from the simplest of them, most of these models have a rather artificial flavor, and do not lead to convincing alternative explanations either of dark matter or of dark energy. In addition, many of them do not lead (contrary to GR) to mathematically “well-posed” evolution problems [84–86]. This entails a serious challenge to deriving strong-field predictions for such models. It has been argued that many of these (dark-energy motivated) models should be viewed as effective field theory (EFT) approximations that need some sort of UV completion at an unusually low frequency scale [87]. In spite of these shortcomings, such models are conceptually interesting because they give examples of deviations for various predictions of GR, existing independently from each other, in various regimes. For instance, some special tensor-scalar models lead to black hole solutions modified by scalar-hair [88–90]. For other types of black holes with scalar-hair, see Ref. [91]. This shows the interest of phenomenologically testing, in a democratic and agnostic way, all conceivable deviations from GR.

Let us now turn to briefly presenting current experimental results of various phenomenological tests of the main GR predictions recalled in Section 21.2 above.

## 21.4 Experimental tests of the Equivalence Principle (i.e., of the matter-gravity coupling)

### 21.4.1 Tests of the constancy of constants

Stringent limits on a possible time variation of the basic coupling constants have been obtained by analyzing a natural fission reactor phenomenon that took place at Oklo, Gabon, two billion years ago [92,93]. These limits are at the $1 \times 10^{-8}$ level for the fractional variation of the fine-structure constant $\alpha_{\text{em}}$ [93], and at the $4 \times 10^{-9}$ level for the fractional variation of the ratio $m_q/\Lambda_{\text{QCD}}$ between the light quark masses and $\Lambda_{\text{QCD}}$ [94]. The determination of the lifetime of Rhenium 187 from isotopic measurements of some meteorites dating back to the formation of the solar system (about 4.6 Gyr ago) yields comparably strong limits [95]. Measurements of absorption lines in astronomical spectra also give stringent limits on the variability of both $\alpha_{\text{em}}$ and $\mu = m_p/m_e$ at cosmological redshifts, e.g.,

$$ \Delta\alpha_{\text{em}}/\alpha_{\text{em}} = (1.2 \pm 1.7)_{\text{stat}} \pm 0.9_{\text{sys}}) \times 10^{-6}, \qquad (21.51) $$

at redshifts $z = 1.0-2.4$ [96], and

$$ |\Delta\mu/\mu| < 4 \times 10^{-7} (95\% \text{ C.L.}), \qquad (21.52) $$

at a redshift $z = 0.88582$ [97]. There are also significant limits on the variation of $\alpha_{\text{em}}$ and $\mu = m_p/m_e$ at redshift $z \sim 10^3$ from cosmic microwave background data, e.g., $\Delta\alpha_{\text{em}}/\alpha_{\text{em}} = (3.6 \pm 3.7) \times 10^{-3}$ [98]. Direct laboratory limits (based on monitoring the frequency ratio of several different atomic clocks) on the present time variation of $\alpha_{\text{em}}$, and $\mu = m_p/m_e$ have reached the levels [99,100]

$$ \begin{aligned} d \ln(\alpha_{\text{em}})/dt &= (1.8 \pm 2.5) \times 10^{-19} \text{yr}^{-1}, \\ d \ln(\mu)/dt &= (-8 \pm 36) \times 10^{-18} \text{yr}^{-1}. \end{aligned} \qquad (21.53) $$

There are also experimental limits on a possible dependence of coupling constants on the gravitational potential [99,100].

Experimental limits on the present time variation of the gravitational constant, Eq. (21.33), have been derived from planetary ephemerides [101], lunar laser ranging [102], and binary-pulsar data [103,104]. The most stringent limits come from lunar-laser-ranging data [102]:

$$ \frac{\dot{G}_0}{G_0} = (7.1 \pm 7.6) \times 10^{-14} \text{ yr}^{-1}. \qquad (21.54) $$

### 21.4.2 Tests of the isotropy of space and of Local Lorentz invariance

The highest precision tests of the isotropy of space have been performed by looking for possible quadrupolar shifts of nuclear energy levels [105]. The (null) results can be interpreted as testing the fact that the various pieces in the matter Lagrangian, Eq. (21.6), are indeed coupled to the same external metric $g_{\mu\nu}$ to the $10^{-29}$ level.

Stringent tests of possible violations of local Lorentz invariance in gravitational interactions have been obtained both from solar-system data [8] and pulsar data [106,107]. For astrophysical constraints on possible Planck-scale violations of Lorentz invariance, see Ref. [108].

### 21.4.3 Tests of the universality of free fall (weak, and strong equivalence principles)

The universality of the acceleration of free fall has been verified, for laboratory bodies, both on the ground [109,110] (at the $10^{-13}$ level), and in space [111,112] (at the $10^{-15}$ level):

$$ \begin{aligned} (\Delta a/a)_{\text{BeTi}} &= (0.3 \pm 1.8) \times 10^{-13}; \\ (\Delta a/a)_{\text{BeAl}} &= (-0.7 \pm 1.3) \times 10^{-13}; \\ (\Delta a/a)_{\text{TiPt}} &= (-1.5 \pm 2.3(\text{stat}) \pm 1.5(\text{syst})) \times 10^{-15}. \end{aligned} \qquad (21.55) $$

The universality of free fall has also been verified when comparing the fall of classical and quantum objects (at the $6 \times 10^{-9}$ level
[113]), or of two quantum objects (at the $(0.3 \pm 5.4) \times 10^{-7}$ [114], and $(1 \pm 1.4) \times 10^{-9}$, levels [115]).

The universality of free fall of self-gravitating bodies (strong equivalence principle) has been verified in both the weak-gravity, and the strong-gravity regimes. The gravitational accelerations of the Earth and the Moon toward the Sun have been checked to agree at the $10^{-13}$ level [102]

$$ (\Delta a/a)_{\text{EarthMoon}} = (-3 \pm 5) \times 10^{-14}. \qquad (21.56) $$

The latter result constrains the Nordtvedt PPN parameter [9] $\eta \equiv 4\bar{\beta} - \bar{\gamma}$ to the $10^{-4}$ level:

$$ \eta = (-0.2 \pm 1.1) \times 10^{-4}. \qquad (21.57) $$

See below for strong-field tests of the strong equivalence principle.

Finally, the universality of the gravitational redshift of clock rates has been verified at the $10^{-4}$ level by comparing a hydrogen-maser clock flying on a rocket up to an altitude of about 10,000 km to a similar clock on the ground [116]. The redshift due to a height change of only 33 cm has been detected by comparing two optical clocks based on $^{27}$Al$^+$ ions [117]. The gravitational redshift has also been detected in the orbit of a star near the supermassive black hole at the center of our Galaxy [118, 119], and its universality has been verified at the 5% level [120].

## 21.5 Tests of quasi-stationary, weak-field gravity

All currently performed gravitational experiments in the solar system, including perihelion advances of planetary orbits, the bending and delay of electromagnetic signals passing near the Sun, and very accurate ranging data to the Moon obtained by laser echoes, are compatible with the post-Newtonian results of Eq. (21.15), Eq. (21.13), and Eq. (21.14). The “gravito-magnetic” interactions $\propto v_A v_B$ contained in Eq. (21.13) are involved in many of these experimental tests. They have been particularly tested in lunar-laser-ranging data [121], in the combined LAGEOS-LARES satellite data [122, 123], and in the dedicated Gravity Probe B mission [124].

To assess in a quantitative manner the results of the various solar-system tests of gravity it is convenient to express them in terms of the PPN parameters defined above. The best current limit on the post-Einstein parameter $\bar{\gamma} \equiv \gamma - 1$ is

$$ \bar{\gamma} = (2.1 \pm 2.3) \times 10^{-5}, \qquad (21.58) $$

as deduced from the additional Doppler shift experienced by radio-wave beams connecting the Earth to the Cassini spacecraft when they passed near the Sun [125].

The (cubic-vertex-related) post-Einstein parameter $\bar{\beta} \equiv \beta - 1$ is constrained at the $10^{-4}$ level both from a study of the global sensitivity of planetary ephemerides to post-Einstein parameters [101],

$$ |\bar{\beta}| < 7 \times 10^{-5}, \qquad (21.59) $$

and from lunar-laser-ranging data [102]

$$ \bar{\beta} = (-4.5 \pm 5.6) \times 10^{-5}. \qquad (21.60) $$

The periastron advance of the star S2 around the Galactic center massive black hole has been observed to agree with GR within 20% [126]. More stringent limits on $\bar{\gamma}$ (i.e. the coupling of $\varphi$ to matter) are obtained in dilaton-like models where scalar couplings violate the Equivalence Principle [127].

## 21.6 Tests of strong-field gravity (neutron stars and black holes)

Experimental tests of strong-field gravity have been obtained in various physical systems, notably binary pulsars and coalescing binary black holes.

It is convenient to quantitatively express binary-pulsar tests of strong-field gravity by using the PPK formalism defined above. We recall that the measurement of N phenomenological PPK parameters leads to N - 2 tests of strong-field gravity. In all, thirteen tests of strong-field and/or radiative gravity have been obtained in the four different (double neutron-star) binary pulsar systems PSR1913+16 [10, 11, 128], PSR1534+12 [129–131],

PSR J1141–6545 [132–135], and PSR J0737–3039 A,B [136–140]. These consist of $N-2 = 5-2 = 3$ tests from PSR1913+16; $5-2 = 3$ tests from PSR1534+12; $4-2 = 2$ tests from PSR J1141–6545; and $7-2 = 5$ tests from PSR J0737–3039 (see, also, Ref. [141] for additional, less accurate tests of relativistic gravity). Among these tests, four of them (those involving the measurement of the PPK parameter $\dot{P}_b$) probe radiative effects, and will be discussed in the following section. The four binary pulsar systems PSR1913+16, PSR1534+12, PSR J1141–6545, and PSR J0737–3039 A,B have given nine tests of quasi-static, strong-field gravity. GR passes all these tests within the measurement accuracy. Let us only highlight here some of the most accurate strong-field tests.

In the binary pulsar PSR 1534+12 [129] one has measured five post-Keplerian parameters: $k, \gamma_{\text{timing}}, r, s$, and (with less accuracy) $\dot{P}_b$ [130, 131]. This yields three tests of relativistic gravity. Among these tests, the two involving the measurements of $k, \gamma_{\text{timing}}, r$, and $s$ accurately probe strong field gravity, without mixing of radiative effects [130]. The most precise ($10^{-3}$ level) of these pure strong-field tests is the one obtained by combining the measurements of $k, \gamma_{\text{timing}},$ and $s$; namely, [131],

$$ \left[ \frac{s^{\text{obs}}}{s^{\text{GR}}[k^{\text{obs}}, \gamma_{\text{timing}}^{\text{obs}}]} \right]_{1534+12} = 1.002 \pm 0.002. \qquad (21.61) $$

The discovery of the remarkable double binary pulsar PSR J0737–3039 A and B [136, 137] has led to the measurement of seven independent parameters [138–140]: five of them are the post-Keplerian parameters $k, \gamma_{\text{timing}}, r, s$, and $\dot{P}_b$ entering the relativistic timing formula of the fast-spinning pulsar PSR J0737–3039 A; a sixth is the ratio $R = x_B/x_A$ between the projected semi-major axis of the more slowly spinning companion pulsar PSR J0737–3039 B, and that of PSR J0737–3039 A (the theoretical prediction for the ratio $R = x_B/x_A$, considered as a function of the (inertial) masses $m_1 = m_A$ and $m_2 = m_B$, is $R^{\text{theory}} = m_1/m_2 + O((v/c)^4)$ [13, 14], independently of the gravitational theory considered). Finally, the seventh parameter $\Omega_{\text{SO,B}}$ is the angular rate of (spin-orbit) precession of PSR J0737–3039 B around the total angular momentum vector [139, 140]. These seven measurements give us five tests of relativistic gravity [138, 142, 143], four of which are quasi-static, strong-field tests. GR passes all those tests with flying colors [144]. The most accurate is at the $2 \times 10^{-4}$ level:

$$ \left[ \frac{s^{\text{obs}}}{s^{\text{GR}}[k^{\text{obs}}, \gamma_{\text{timing}}^{\text{obs}}]} \right]_{0737-3039} = 1.00009 \pm 0.00018. \qquad (21.62) $$

Binary pulsar data on other types of pulsar systems can be used to test strong-field aspects of the “strong equivalence principle,” namely the GR prediction that strong-self-gravity objects (such as neutron stars) should fall with the same acceleration as weak-self-gravity objects (such as white-dwarfs) in the (external) gravitational field created by other objects (such as the Galaxy, or another white dwarf). The first binary-pulsar tests of this property have been obtained in nearly circular binary systems (made of a neutron star and a white dwarf) falling in the field of the Galaxy, and have led to strong-field confirmations (at the $2 \times 10^{-3}$ level) of the strong equivalence principle [104, 145–147]. The remarkable discovery of the pulsar PSR J0337+1715 in a hierarchical triple system [148] has allowed one to derive a much more accurate test of the strong equivalence principle because the inner binary (comprising a pulsar and a close white-dwarf companion) falls toward the outer white-dwarf companion with an acceleration that is $10^8$ times larger than the Galactic acceleration. This leads to a 95% confidence level limit on a possible fractional difference in free-fall acceleration of the pulsar and its close companion of [149, 150]

$$ |\Delta a/a| < 2.05 \times 10^{-6} (95\% \text{ C.L.}). \qquad (21.63) $$

This limit yields strong constraints on tensor-scalar gravity models.

Measurements over several years of the pulse profiles of various pulsars have detected secular changes compatible with the prediction [151] that the general relativistic spin-orbit coupling should
cause a secular change in the orientation of the pulsar beam with respect to the line of sight ("geodetic precession"). Such confirmations of general-relativistic spin-orbit effects were obtained in PSR 1913+16 [152], PSR B1534+12 [131], PSR J1141-6545 [153], PSR J0737-3039 [139, 140], and PSR J1906+0746 [154, 155]. In some cases (notably PSR 1913+16 and PSR J1906+0746) the secular change in the orientation of the pulsar beam is expected to lead to the disappearance of the beam (as seen on the Earth) on a human time scale (the second pulsar in the double system PSR J0737-3039 already disappeared in March 2008 and is expected to reappear around 2035 [140]).

Recently, the ultimate strong-field regime of black holes has started to be quantitatively probed via GW observations. The LIGO-Virgo(-Kagra) collaboration has detected (starting in September 2015) GW signals [35], which, besides testing the radiative structure of gravity (see next section), are in excellent qualitative and quantitative agreement with the structure and dynamics of black-hole horizons in GR. Because of the mixing of strong-field effects with radiative effects during the coalescence of two black holes, and because of the lack of detailed alternative-theory predictions for this process (see, however, Refs. [46-48]), it is not easy to set quantitative limits on possible strong-field deviations from GR, independently of radiative effects. Direct tests of the existence of black-hole horizons are scarce (see, however, Sec. VIIIIB of [156] which reports the lack of any statistical evidence for GW echoes).

Let us also mention that the Event Horizon Telescope collaboration has obtained event-horizon-scale images of the supermassive black hole candidates in the center of the giant elliptical galaxy M87, and in the center of our Galaxy. These images are consistent with GR models of accreting Kerr black holes [157, 158]. For discussions of the corresponding constraints on the black hole geometry in the vicinity of the light ring see Refs. [159-161].

## 21.7 Tests of radiative gravity (both in binary-pulsar data and in GW data)

Experimental confirmations of the GR predictions for the radiative structure of gravity have been obtained both in binary-pulsar data and in the observation of GW signals from coalescing compact binaries (binary black holes and binary neutron stars).

Binary-pulsar observations involving the measurement of the orbital period derivative $\dot{P}_b$ give direct experimental tests of the reality of gravitational radiation, and, in particular, an experimental confirmation that the speed of propagation of gravity $c_g$ is equal to the speed of light $c$ (indeed, as recalled above, $\dot{P}_b$ is a consequence of the propagation of the gravitational interaction between the two neutron stars [12]). Even in the presence of screening mechanisms within the binary system, the value of $\dot{P}_b$ yields a measurement of the speed of propagation of GWs at the $10^{-2}$ level [162]. The currently most accurate binary-pulsar tests of the radiative properties of gravity come from the binary neutron-star systems PSR1913+16 and PSR J0737-3039 A,B, as well as from several neutron-star-white-dwarf systems, notably PSR J1738+0333.

After subtracting a small ($\sim 10^{-14}$ level in $\dot{P}_b^{\text{obs}} = (-2.423 \pm 0.001) \times 10^{-12}$), but significant, “Galactic” perturbing effect (linked to Galactic accelerations and to the pulsar proper motion) [163], one finds that the phenomenological test obtained by combining the measurements of the three PPK parameters ($k - \gamma_{\text{timing}} - \dot{P}_b$) is passed by GR with complete success [128]:

$$ \left[ \frac{\dot{P}_b^{\text{obs}} - \dot{P}_b^{\text{gal}}}{\dot{P}_b^{\text{GR}}[k^{\text{obs}}, \gamma_{\text{timing}}^{\text{obs}}} \right]_{1913+16} = 0.9983 \pm 0.0016. \quad (21.64) $$

Here $\dot{P}_b^{\text{GR}}[k^{\text{obs}}, \gamma_{\text{timing}}^{\text{obs}}]$ is the result of inserting in $\dot{P}_b^{\text{GR}}(m_1, m_2)$ the values of the masses predicted by the two equations $k^{\text{obs}} = k^{\text{GR}}(m_1, m_2)$, and $\gamma_{\text{timing}}^{\text{obs}} = \gamma_{\text{timing}}^{\text{GR}}(m_1, m_2)$. This yields experimental evidence for the reality of gravitational radiation damping forces at the $(-1.7 \pm 1.6) \times 10^{-3}$ level.

An even better experimental test of the radiative structure of gravity ($6 \times 10^{-5}$ level) has been recently obtained from the com-

bined measurement in PSR J0737-3039 A,B of the three parameters $k, s$, and $\dot{P}_b$ [144]:

$$ \left[ \frac{\dot{P}_b^{\text{obs}} - \dot{P}_b^{\text{gal}}}{\dot{P}_b^{\text{GR}}[k^{\text{obs}}, s^{\text{obs}}]} \right]_{0737-3039} = 0.999963 \pm 0.000063. \quad (21.65) $$

In addition to the above tests, further very stringent tests of radiative gravity follow from the measurement of the orbital period decay $\dot{P}_b$ of low-eccentricity pulsar-white dwarf systems. Notably, the system PSR J1738+0333 yields an intrinsic orbital decay of [164]

$$ \left[ \dot{P}_b^{\text{obs}} - \dot{P}_b^{\text{gal}} \right]_{1738+0333} = (-25.9 \pm 3.2) \times 10^{-15}, \quad (21.66) $$

to be compared to

$$ \left[ \dot{P}_b^{\text{GR}} \right]_{1738+0333} = (-27.7^{+1.5}_{-1.9}) \times 10^{-15}. \quad (21.67) $$

The fractional agreement between the (corrected) observed period decay and the GR-predicted one seems to be quantitatively less impressive than the double-neutron-star results cited above, but the crucial point is that asymmetric binary systems (such as neutron-star-white-dwarf ones) are strong emitters of dipolar gravitational radiation in tensor-scalar theories, with $\dot{P}_b$ scaling (modulo matter-scalar couplings) like $m_1 m_2 / (m_1 + m_2)^2 (v/c)^3$, instead of the parametrically smaller GR-predicted quadrupolar radiation $\dot{P}_b \sim (v/c)^5$ [7, 15]. In view of the very small absolute value of $\dot{P}_b$, this makes such systems (and notably PSR J1738+0333) very sensitive probes of tensor-scalar gravity [103, 164-168]. It is then useful to turn to a theory-dependent analysis of pulsar data. Such an analysis (see, e.g., [17, 130, 164, 167, 168]) leads to excluding a large portion of the parameter space of tensor-scalar gravity allowed by solar-system tests. As a result, the basic matter-scalar coupling $\alpha_0^2$ is more strongly constrained, over most of the parameter space, than the best current solar-system limits of Eq. (21.58) (namely below the $10^{-5}$ level) [164, 167].

We now turn to the tests of radiative gravity that can be deduced from the GW data gathered from the first three observing runs of the LIGO-Virgo-Kagra collaboration. All currently detected GW signals are consistent with GR predictions. Several phenomenological approaches were used and led to setting limits on possible deviations from GR [51, 156, 169, 170].

A theory-agnostic quantitative assessment on possible deviations from GR is given by measuring the agreement between the full observed GW signal of coalescing binary black holes, and the GR-predicted one. One of the strongest results was obtained with the first event: GW150914, which had an SNR of 24. The noise-weighted correlation between the GW150914 signal and the best-fit GR-predicted waveform was found to be $\ge 97\%$ [51, 169]. In other words, GR-violation effects that cannot be reabsorbed in a redefinition of physical parameters are limited (in a noise-weighted sense) to less than 3%.

Besides checking the agreement between the full observed GW signals and the corresponding best-fit full signals predicted by GR, one also tested the consistency between separate parts of the signals. A first approach [156, 169, 170] separates: (i) the lower-frequency signal emitted during the inspiral phase (considered up to the innermost stable circular orbit); and (ii) the higher-frequency remaining signal emitted during the postinspiral phase, comprising the late-inspiral, the merger, and the ringdown. Separately fitting each of these partial signals to GR-based templates then yields separate estimates of the binary's parameters, leading to separate estimates of the mass $M_f$ and dimensionless spin parameter $a_f = J_f / (GM_f^2)$ of the final black hole that would be formed (in GR) by the coalescence of the two initial black holes. The consistency with GR then consists in testing whether the two estimates ($M_f, a_f$)$_{\text{insp}}$ and ($M_f, a_f$)$_{\text{postinsp}}$ are compatible with each other. They were found to be compatible for all events whose corresponding separate SNRs made such an analysis meaningful (see Figs. 3 and 4 in [170] and Fig.3 in [156]). Quantitatively, the
(population-marginalized) fractional differences

$$ \begin{aligned} \frac{\Delta M_f}{\bar{M}_f} &= 2 \frac{M_f^{\text{insp}} - M_f^{\text{postinsp}}}{M_f^{\text{insp}} + M_f^{\text{postinsp}}}, \\ \frac{\Delta a_f}{\bar{a}_f} &= 2 \frac{a_f^{\text{insp}} - a_f^{\text{postinsp}}}{a_f^{\text{insp}} + a_f^{\text{postinsp}}}, \end{aligned} \quad (21.68) $$

between the two estimates were found to be consistent with zero (i.e. with GR) [156]

$$ \begin{aligned} \frac{\Delta M_f}{\bar{M}_f} &= -0.02^{+0.07}_{-0.06}, \\ \frac{\Delta a_f}{\bar{a}_f} &= -0.06^{+0.10}_{-0.07}. \end{aligned} \quad (21.69) $$

A second approach studies the consistency of the ringdown GW signal (emitted by the final, vibrating black hole) with GR predictions. See, in particular, section VIII A, and Figs. 13 and 14, of [156]. The results are rather sensitive to various data analysis assumptions (notably the use of complete waveform models versus an analysis using only the ringdown signal). Though there is no clear sign of any deviations from GR, the current SNRs do not yield strong theory-agnostic evidence for the presence of ringing overtones.

The parametrization of Eq. (21.38) for possible deviations in the frequency dependence of the Fourier-domain phase $\psi(f)$ of the black hole coalescence GW signal was used to measure best-fit values for each fractional deviation parameter $\delta\hat{p}_i$, considered separately (the other ones being set to zero). In all cases, the posterior distribution for each $\delta\hat{p}_i$ is consistent with the GR value, i.e., $\delta\hat{p}_i^{\text{GR}} = 0$ (see Fig. 6 in [156]). The current limits on $\delta\hat{p}_i$ are (roughly) of order unity, except for the two parameters highlighted above: $\delta\hat{p}_3$ (parameterizing the $O((\frac{v}{c})^3)$ fractional correction to the LO, quadrupolar term); and $\delta\hat{p}_{-2}$ (parameterizing a possible dipolar-radiation-related $O((\frac{v}{c})^{-2})$ fractional correction to the LO, quadrupolar term). Ref. [170] gives 90%-credible intervals for $\delta\hat{p}_3$ of $-0.02^{+0.11}_{-0.10}$ (when using a phenomenological model [53]), and $-0.01^{+0.10}_{-0.11}$ (when using an effective one-body model [38]). Ref. [156] gives a 90%-credible upper bound for a possible dipolar term of $|\delta\hat{p}_{-2}| \le 7.3 \times 10^{-4}$.

As recalled above, GR predicts that the polarization content of GWs is pure helicity-2, i.e. described by the two independent components of a traceless tensor transverse to the propagation direction. A (massless) scalar excitation would add a pure-trace “breathing mode” in the plane transverse to the propagation direction. A phenomenological approach to generic metric theories of gravity would allow for up to six polarizations for a GW [171], namely two tensor, two vector and two scalar modes. The LVK collaboration tested possible polarization deviations from GR in various ways [33, 156, 169]. Recent results are given in section VII of [156]. Though pure-scalar, pure-vector and vector-scalar hypotheses are significantly disfavored, any mixed hypothesis involving tensor modes (i.e., tensor-scalar, tensor-vector, and tensor-vector-scalar) cannot be ruled out conclusively.

GR also predicts that GWs are non dispersive, and propagate at the same speed as light. One can phenomenologically modify the GR-predicted GW phase evolution by adding the putative effect of an anomalous dispersion relation of the form $E^2 = p^2c^2 + Ap^\alpha c^\alpha$. GW data have been used to set bounds on the anomalous coefficient A for various values of the exponent $\alpha$. The case $\alpha = 0$ is equivalent to assuming that gravitons disperse as a massive particle [59]. The current (90%-credibility) phenomenological GW limit on the graviton mass is $m_g \le 1.27 \times 10^{-23} \text{eV/c}^2$ [156]. This is 2.5 times more stringent than the Solar System bound of $3.16 \times 10^{-23} \text{eV/c}^2$ [65].

Finally, a very constraining bound on the speed of propagation of gravity $c_{\text{GW}}$ was derived from the observed time delay of 1.7 s between GW170817 and the associated $\gamma$-ray burst. Namely, the fractional difference between $c_{\text{GW}}$ and $c_{\text{light}} \equiv c$ is constrained to

be [172]

$$ -3 \times 10^{-15} < \frac{c_{\text{GW}} - c}{c} < +7 \times 10^{-16}. \quad (21.70) $$

When comparing the latter bound to the prediction Eq. (21.50) from general second-order tensor-scalar theories, Eq. (21.49), one is led to conclude that the coupling function $G_5(\varphi, X)$ has to be ignored and that the coupling function $G_4(\varphi, X)$ has to be restricted to depend only on $\varphi$. This drastically reduces the viable tensor-scalar modified-gravity models [173–176].

Let us finally mention that four pulsar timing arrays have recently given tantalizing evidence for the existence of a stochastic background of gravitational waves with frequency $\sim 10^{-9}$ Hz [177–180].

## 21.8 Conclusions

All present experimental tests are compatible with the predictions of the current “standard” theory of gravitation, Einstein’s General Relativity. Let us recap the main tests. The universality of the coupling between matter and gravity (Equivalence Principle) has been verified at around the $10^{-15}$ level. Solar system experiments have tested the weak-field predictions of Einstein’s theory at the few times $10^{-5}$ level. The propagation properties (in the near zone) of relativistic gravity, as well as several of its static strong-field aspects, have been verified at the $10^{-4}$ level (or better) in several binary pulsar experiments. Interferometric detectors of gravitational radiation have given direct observational proofs of the existence, and properties, of gravitational waves (in the wave zone), and of the existence of coalescing black holes, and they have already set strong limits on possible deviations; in particular: an upper bound $|\delta\hat{p}_{-2}| < 7.3 \times 10^{-4}$ on a possible dipolar contribution to the GW flux; the $O(10^{-15})$ bound of Eq. (21.70) on the speed of gravity; and confirmation of the tensor polarization structure of gravitational waves. In addition, laboratory experiments have set strong constraints on sub-millimeter modifications of Newtonian gravity, while many different cosmological data sets have been used to set limits on possible GR deviations on cosmological scales [29]. In spite of the uneasiness of having to assume the existence of dark matter, and the presence of an unnaturally small cosmological constant (as dark energy), General Relativity stands out as a uniquely successful description of gravity on all the scales that have been explored so far. There are no modified-gravity models which naturally pass all existing experimental tests, while either explaining away the need for dark matter or for dark energy.

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